Mätam, the term used in South Asia for the act of self-flagellation during the Shia remembrance of Muharram. Ma'tam (مأتم), a Shia congregation hall in Bahrain, known as a Hussainia or Imambargah elsewhere.
The main point is that you believe and feel grief for Imam Husain (AS). It is a condition for every believer to love the Prophet (SAWA) and his Holy Progeny Ahlul Bayt (AS) more than loving himself and his own family.
The way how to express this noble feeling of grief and sadness depends on you and your culture but it must be within the frame of Islamic rules
You were brought up in USA, so you may not be familiar with the Matam practiced by your brothers in faith who came from the Indian sub continent or other countries. You wrote that you felt wrong on seeing them. This feeling does not make you away from following Ahlul Bayt (AS) as far as you believe in Ahlul Bayt (AS) being the Most pious leaders of Islam after the Prophet Muhammad (SAWA) and you feel grief for their tragedies.
Islam as the religion for all human beings, gives all of them the choice to express their feelings according to their own different cultures and ways as far as it remains in the frame of Shariah.
Yes, you can still be considered a true follower of Ahl al-Bayt if you do not take part in matam or public mourning ceremonies. These things are optional. The most important things are (a) inner belief (i.e. belief in the theology taught by Ahl al-Bayt as well as belief in their authority), and (b) following them in your outer actions to the best of your ability (acts of worship, how you treat others, how you live, etc).
Of course, as you are likely aware, one of the things that is mentioned in hadith is that the followers of Ahl al-Bayt feel happy at their times of happiness and sad in their times of sadness; that is, there is a sort of empathy or emotional link. Since you say you feel grief, this is already there; I am just mentioning it so it is not neglected.
I agree that sometimes people from more reserved cultures are uncomfortable with matam ceremonies. This is particularly the case if someone grew up with the tacit message that expressions of emotion are socially unacceptable, unmanly, weak, undignified, etc, or if someone was punished for them.
Sometimes people from a Sunni background are also uncomfortable at these gatherings (even if they come from emotionally expressive cultures).
However, even if you choose not to participate in these activities, it is good to acknowledge and respect that many other Shi'is do and this is the way they express their emotions and loyalty towards Ahl al-Bayt. That is, it is better simply to acknowledge that it is one's personal preference not to attend, rather than to try to make a blanket statement that it is wrong for others to do so. There is a strong spiritual component to these gatherings (although I could understand that this might not be felt if one is feeling shock instead), and they do function to forge a link between the individual and the teachings of Ahl al-Bayt that can come into play in other life circumstances.
To some degree, you will miss out on a sense of community spirit, belonging, or shared experience by not participating in these activities, because they are so widespread, but this is a different issue.
Also, this may or may not be of interest, but if you do look around at world religions, there are actually a lot of religions that have ritual or spiritual acts which involve a sort of emotional/intellectual abandon or self-harm. (For instance, speaking in tongues or nailing one's self to a cross) What makes these things "safe" ways of exploring or expressing one's spirituality is that they are controlled and there are unwritten rules about what is and is not acceptable, and when. For instance, someone walking down the street randomly doing matam would be seen as mentally unstable, but someone doing it in a ritual setting at the appropriate time would be seen as normal. Also this is similar for a some Sufi practices. This is more of a comparative religious studies perspective, but I just thought I'd put it out there.
Anyway, back to the main question, here are some hadith (which you may have already read!) about what constitutes a true follower of Ahl al-Bayt:
Imam al-Hasan (a.s.) said in answer to a man who said to him, ‘Verily I am one of your Shi’ah’, ‘O ‘Aabdallah, if you are truly obedient to us in our commands and prohibitions, then you are telling the truth. But if not, then do not add to your sins by falsely claiming such a dignified position that you are not worthy of. Do not say, ‘I am one of your Shi’ah’, but say rather, ‘I am one of your adherents and one of your lovers and an enemy to your enemies.’ You are [doing] good and aiming towards good.’[Tanbih al-Khawatir, v. 2, p. 106]
Imam al-Baqir (a.s.) said, ‘Our Shi’ah are none other than those who are consciously wary of their duty to Allah and obey Him. They are known solely for their humbleness, their humility, their returning promptly whatever is entrusted in their care and their Abundant remembrance of Allah.’[Tuhaf al-’Uqul, p. 295]
Imam al-Sadiq (a.s.) said, ‘Verily the Shi’ah of Ali were those who restrained their stomachs and their sexual desires, who struggled and fought intensely, who worked hard for their Creator, who hoped for His reward and feared His punishment. If you have seen such people, then they are the very Shi’ah of Ja’afar.’[al-Kafi, v. 2, p. 233, no. 9]
Imam al-Sadiq (a.s.) said, ‘Test our Shi’ah with regard to three things: the prayer times to see how well they observe them, their secrets to see how well they guard them from our enemies, and their wealth to see how they help out their fellow brothers with it.’[Bihar al-Anwar, v. 83, p. 22, no. 40]
Anyway, I hope you are able to find a way to sort out the unease you are feeling.