Often, the production of hair oil adjusts naturally over a few months to a change in hair routine. So, possibly, your hair will adjust by itself.

In the meantime, I suggest using the minimum water and doing the minimum necessary amount of washing for wudu (rather than anything extra/mustahhab).

This will vary between schools of thought. For instance, the 4 Sunni schools of thought require wiping the entire head (but have some minor differences regarding wudu between each other). The Twelver Shi'i school of thought requires wiping a quarter of the head with 1-3 fingers by using the remaining water from wudu (not taking extra water). 

Historically, most people did not wash their hair daily, and certainly did not have access to commercially produced shampoos, so, in and of itself, performing wudu regularly does not usually require people to wash their hair more frequently (although individual circumstances vary).

This is venturing outside the scope of responses offered here, but you could look into advice online by people starting the "no shampoo (no 'poo)" routine in reducing oiliness. Possibly dry shampoo might be helpful. Of course you have to decide what is best for yourself, your health, and your circumstances.

Lastly... doing wudu doesn't usually make people's hair oily. It is possible that this is happening for yourself, since everyone is different. However, if you haven't already, you could look into other causes, such as hormones, stress, diet, or an underlying health condition. If you have recently converted to Islam, this could be an underlying stressor, if it has come with shifts in your life or challenges.

Best wishes!

Qamar dar Aqrab refers to the moon being in the zodiacal sign of Scorpio which happens 2-3 days per month (in that the moon circulates through all the zodiacal signs in one lunar month) and can be determined via astronomical software or eyeballs.

Historically, in a number of cultures, this has been considered an unfavourable time, which is probably why it appears in our heritage. There are some narrations on not marrying or starting a journey at this time, but they are not considered strong.

Some people choose to avoid scheduling marriages at this time, and that is fine. It is also fine not to pay attention to it at all. People vary in their belief in these things.

However, it is good not to get too caught up in these things, e.g. if someone gets married at this time, it doesn't mean that it will be a disaster. Sometimes if we feel that something has a bad "omen", it becomes a self-fulfilling prophecy because then we subconsciously do things to make it go badly, and that isn't healthy at all! 

If something happens by accident (for instance, conception), there is no need to worry. Allah knows best what He creates and when, and He has his wisdom for it, especially when it comes to bringing life into this world, which is in His hands.

Of course, it doesn't hurt to give sadaqa a few times if you are concerned, but it isn't necessary. Don't overburden yourself with it by making it a requirement!

In general, most of the acts of hajj are similar for men and women, apart from rulings related to women's issues (which I'm sure you're familiar with) and a few requirements of the hajj that are more lenient for women, which is particularly helpful these days because of the crowding.

However, beyond that, I think some things that might pertain to women are, firstly, what to wear. Although the clothing requirements are less strict for women, may women wear an all-white garment; if possible, I would suggest a thin and natural fabric (although not see-through) in the warmer months, especially if you are coming from a cooler climate. Polyester gets quite hot very quickly. If you follow the Twelver Shi'i school of thought, you should also avoid non-halal-leather shoes due to concerns about najasa. 

Second, one unfortunate reality is that not everyone who is present in Mecca during the hajj time is there for pious reasons and is of good character, and sexual harassment, inappropriate touching, or safety threats are known to happen. So, it is good to use the same precautions that one might use in travelling in any other new place (for instance, avoiding going places alone unless it seems safe).

Third, overall, people are difficult, and other women are no exception. I have found that the companionship one has during a religious journey often greatly affects the journey. Some people are absolutely lovely and greatly enhance the experiences as models of good ethics and faith. However, not everyone who actually attends hajj is there with a mindset that they are primarily making a journey to Allah. And even those who do have that thought may still suffer from everyday human failings. The pressures of the hajj journey can bring out the worst in people.

This is true for both men and women. However, since women often deal with women, we can talk about women here. I have noticed that some women who have had very restricted lives, who are unused to cultures other than their own, and/or who are used to having the same everyday habits (such as mealtimes) sometimes have a hard time with the hajj and take this out on others. (By this, I am not referring to women from villages, who also tend to be lovely.) Among women, this is sometimes expressed in the types of behavior that is stereotypically associated with women, such as being gossipy, arguing, making judgmental/hurtful comments about clothing or other habits different from their own cultural expectations, etc. This is apart from the fact that many people perform hajj at an older age, and, these days, many people at an older age suffer from health conditions such as blood sugar irregularities which can also give them a shorter fuse.

So if that ends up being the case, it is good to try to stay above that and try to stay out of it, and to be patient and forgiving - it is probably meant to be part of the process.

If you are travelling with people who are meaningful to you, such as extended family or in-laws, on the one hand this can provide support and be a nice shared experience, but sometimes interpersonal dynamics can take center stage and distract from focusing on Allah, so it is good to try to keep a balance and not be too focused on other people around, even if hajj is a communal experience. 

Fourth, there is often a lot of time spent shopping for souvenirs. Sometimes, women are tasked with doing this. I understand the desire (or pressure) to bring back souvenirs, and shopping can also be an interesting way to get to know a new place and meet people in another environment, and of course it benefits shopkeepers. Sometimes it is also an opportunity to buy items hard to find back home such as abayas. So it is fine to do what is necessary, but I think it is good to try to spend as little time as possible on this to focus more on the spiritual, historical, or communal aspects of the hajj, and, if buying souvenirs, to stick to things which are spiritually meaningful and straightforward (such as dates from Medina, prayer rugs, and so forth) and avoid mass-produced items which are readily available in many countries. 

Fifth, some people today (men and women) treat the hajj as a sort of holiday, especially if they have a nice hotel and meal service. I have heard some women who do a lot of housework describe it as a holiday from things like cooking and cleaning, and while I am sympathetic to that, I think it is better to avoid thinking of the hajj as a holiday and rather focus on the journey to Allah. Of course it is a blessing to have nice accommodations or provisions, if one has them, and it is helpful to have time off from housework to focus on prayer and duas. These things can make the journey physically easier, if one has them; of course, it is also good to appreciate that many pilgrims do not have them and are in rougher conditions.  (Human nature being what it is, I am not sure that everyone always thinks about these things.) This is just my view of course, and I do not want to come across as judgmental to other people's circumstances. 

Anyway, here is a guidebook for women which might be of interest and which contains details about some of the technicalities: https://www.al-islam.org/guidebook-women-rites-hajj-razia-batool-najafi

May your journey be very blessed!

I am not aware of any narrations on this matter, although it is something you could deduce through historical records and calendrical calculations (and others have done this in the past - you can search online for the results).

However, there is a narration between Imam al-Sadiq (A) and an astrologer in which Imam al-Sadiq (A) associates Imam 'Ali and the awliya' with Saturn. (This is not a zodiacal sign, obviously, but is thematically related.) You can read it here (part 12, number 69): https://www.al-islam.org/al-khisal-numeric-classification-traditions-cha...