Don't go to the psychiatrist that your cousin works for. This is asking for trouble for you, honestly - it's a "broken frame" (assuming the psych is doing some counseling, rather than just prescribing meds?)
In therapy, you have a "frame" for the relationship - which is supposed to make you feel safe enough to be open and honest and share yourself. Part of that is your therapist not knowing the other people in your life, part of it is boundaries around your time (not going over, not kicking you out early), part is not having intrusions during your time (phones ringing, knocks on the door).
Knowing your cousin is right outside the door working, seeing you every time you go in or come out, seeing the doc's reaction after you leave (even if the doc never gives out personal information - what if she looks frustrated or upset or relieved that you've left) - I think this would make the relationship a mess (very unsafe feeling) for most people.
Someone in the comments mentioned that you have borderline issues as well (sorry if I missed that in your post) - which also makes it a bad idea. Won't you be tempted to talk to you cousin, ask her all about your doctor, and find out things you may not want to know?
I'm just thinking that it won't be the best relationship for you, which means you'll end up wasting time and money (and they're so expensive as it is!).
Like everyone else, I think it would be FANTASTIC if you could actually talk to your current psy about what happened. Right now, you are acting completely on your interpretation of events - which could be wrong. How can you know that she's angry (assuming she hasn't told you)?
She's the professional, she should be able to handle it, but you're right - maybe she can't. You need to know FOR SURE before you just ditch her and run away
(speaking as one who's rather an expert at running away, I assure you!)
It's confrontation, and it's scary, because she COULD be totally rejecting. She COULD say, "yes, you really screwed up and upset me and I don't want you to come back."
Hopefully she won't, but sometimes people surprise us in bad ways
And really, when things like that happen, it's often more about THEM then about YOU.
Here's what I'd do. Go to your next appointment. Sit down and tell her, "I've got something really important I need to talk to you about." (If you feel scared/nervous/whatever, you can throw that in too - "but I'm a little nervous about bringing it up...")
Then tell her. Tell her that you realize you said some really hurtful things (or however you'd describe them), if you feel the need to apologize, do that, and then ask - are you upset? Because you've seemed kind of upset to me, and I was wondering if perhaps I should start looking for a new therapist.
She also has a professional obligation to provide references to you, if I understand correctly, so even if she does ask you to go - she may be able to refer you to someone who she thinks will be a better fit.
If you do go, yes, tell her. This way you have a chance for some closure.
Again, this is all assuming she's doing some kind of counseling and you've bonded with her... if she's just giving you prescriptions, I'm not sure how that changes the whole thing.
Best of luck to you, it sounds like a tough situation... and I hope you're feeling a bit more even-keeled now...