Silat al-Rahm

The minimum requirements for silat al-rahm are not too extensive, and there is no shariah requirement that you attend specific gatherings. So it seems like, perhaps, you are more concerned about not wanting to offend your relatives by not attending gatherings; or, perhaps, they are pressuring you. (Or perhaps you are afraid of God, in which case, God created you, and God understands how you feel.) In any case, it seems that what is necessary is to find a balance between your personal needs and their social expectations.

If these are your blood relatives (not adopted relatives, in-laws, etc), it would not be surprising if some of them feel the same way, since personality traits tend to run in the family. So maybe some of them might understand how you feel and also have some suggestions. For instance, there is no reason why, according to shariah, you can't have smaller visits and keep up with your relatives individually, once in a while, on a one-to-one basis, instead of in a big group; or on the telephone/internet, rather than at large gatherings. This might be more comfortable; although this may not fit their preferences.

Tendencies towards introversion or extroversion appear to be inborn and lifelong. However, one of the tasks of adulthood is to develop our opposite, less dominant traits. So, for an introvert, this is developing more facility with socializing or "people-y" situations. 

(I can sympathize as I am also very strongly on the introvert side!)

If you haven't already, you could try reading some of the books available today about socializing for introverts and introvert psychology, since there might be some useful tips about managing busy social situations, recovering your own personal energy after being around too many people, finding your own boundaries, ways to tell people that you need time to yourself, and so forth. 

In my view, due to the spread of digital communications, it is more common these days for people to feel awkward in in-person social gatherings, regardless of whether they are introverts or extroverts, so you could also look at things that have been written about social anxiety or awkwardness. (I am NOT saying you are suffering from this, just saying that it seems to be more common, and there might be something useful.)

Similarly, it's good to remember that other people also sometimes feel awkward in social situations and just don't say it.

Inshallah I hope you find some solutions that work for you!

It is good to treat him/her with respect and maintain some form of communication or offer assistance if it is necessary. However, silat al-rahm (maintaining family ties) is a rather broad concept which can be done in many ways. It is not necessary to have a close or frequent relationship. You can weigh what is appropriate for your situation. In some cases, the use of technology (such as e-mail) can help in being a safe and controlled way to manage the situation. 

The primary thing that the Qur'an empahsises is respect and secondarily compassion. It is also related in hadith that if a person is dealing with parents who beat them, it is best to say "May Allah forgive you". That is, it is good to try to let go of resentment and move towards forgiveness in cases where the parents were not ideal.

For some people, one of the tasks of adulthood is to come to terms with the fact that parents are flawed human beings, like everyone else. We are usually more accepting of strangers with major flaws, but it can be difficult to accept this in a parent, because a child looks to a parent as a godlike figure and provider. The shifting power dynamic in adulthood can help with this, since once someone is no longer in need of their parent for their survival, it is easier to be more objective and accept things as they are. 

Generally, as we progress through adulthood, it becomes easier to recognize and sidestep unhealthy psychological behavior (for instance, being pushed to enable an addiction, relive someone else's trauma, and so forth). Of course you should not participate in haraam activities; generally, as we get older, it gets easier to draw personal boundaries and assert that there are some things we will not do. 

Whether we like it or not, our biological parents are our biological parents, we share some things with them, and there will always be a link. Cutting off ties does not change that. It may suppress the situation or make it no longer a practical concern and may result in more peace of mind, but the link and impact still persists psychologically. 

Also, if you have children, it is healthy for them to have some contact with their grandparents. True, some people do keep children away from their grandparents if they feel they are a bad influence (you mention haraam activities), and this also has to be weighed; but keeping them fully in the dark about their ancestors leaves a blank spot in their minds and sense of self. It also increases the likelihood that the same pattern will be enacted by them (even if you are a good parent towards them). They will subconsciously copy your actions. Similarly, if you speak respectfully of your parents (even if you may feel anger), they will learn that attitude towards parents, but if you speak angrily and critically of them, they will learn that attitude towards parents.

Anyway, life is challenging, and the world is far from ideal. Perhaps the best way to handle the situation for now is simply to pray sincerely for guidance and to see where that takes you.