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!