In a nutshell, it sounds as though your familial interactions are a great source of anxiety for you.
I'm not saying this to be glib, but maybe it would be good for you to go and talk to someone about all this. I grew up with a parent who needed 'taking care of' in the way you describe: emotionally needy, put pressure on me to make her feel better when she was upset, sought out the kids to comfort her and be her companions instead of other adult relationships-- mainly because she was too emotionally crippled to maintain friendships with other adults. It all had to be about HER, and taking care of HER first.
This is one facet of my mother's mental illness (I am not going to go into details or the diagnosis, but stay with how it relates to you). Listen-- when we are doing all of this emotional caregiving for our parents, we are robbed of receiving a 'normal' parenting, including the nuturing and love for "just being" from our parents. We lose the security that children should have in their parents-- that they can protect and care for us as children. Instead, your parents both pressure you and rely on you to be responsible for your mother's emotions.
No wonder you don't feel like being around your family! You feel responsible for their feelings and if they aren't happy, it's your fault! No one would willingly seek out that sort of burden- at least, no one who isn't codependent or nuts. :)
I strongly suggest finding a counselor or therapist to address this with. Talking with them, you will have a good sounding board and hopefully, you will find ways to get healthy within those relationships-- or at least, you can learn to respond to that sense of your family 'needing' you in ways which are healthy for you. Right now, it seems you are afraid to talk about this anger and are having a profound anxiety response, which is pretty reasonable, considering the extreme burden placed on you.
Because I don't know your family, I don't know what else to say. Hopefully, a counselor can help you separate, can help you perceive how much involvement is 'healthy' and as time goes on, you learn how much to give and how much to let go within that relationship. This : " again they accuse me of avoiding, sending curt emails and not addressing an issue that I was clearly angry about. I agree, I have totally done this."-- to me, this sounds like you are so afraid of actually expressing your anger and the consequence of it (that you might upset someone, and this is difficult for those who have been taught that they are responsible for the emotions of others) that the passive-aggressive behavior manifests. You are aware of it and it's sounds like you are also pretty unhappy with that as well.
I send you much encouragement. As someone who has had to learn how to make new boundaries and expectations with my own parents, I do appreciate how hard the work is~~ however, it is essential that we do it so that we can continue going forward in creating healthy relationship dynamics with our own kids. Sometimes, my six year old son comes up to me when I am upset about something and will say something like "I'm trying to make you happy" ...my reply to him is always a hug, a thanks, and a reminder that it's not HIS job to help me be happy. It is his job to listen to the adults, do his homework and his chores, but it is MY JOB to make sure I am happy, not his. I don't want that burden placed on him any more than I wanted it myself, when I was solely responsible for *everything* that happened in my own family of origin.
Good luck going forward. Wanting to change is the first part and often the hardest part (usually denial keeps us stuck)... you can do this! And know that the anger IS a very useful signifier: if you get a chance, check out Harriet Lerner's "Dance of Anger". It's difficult for people who haven't had this experience to understand how emotionally draining and scary it is to have a parent dependent upon you entirely for their happiness. I get it.