firstly it's good that it seems you've finally found some advice on something which sounds like its been going on for a while.
The first bit i'll answer is about the antidepressants, as a person who worked in a diagnostic laboratory is studying a degree in biomedical science and myself has mild liver problems I can tell you that anti depressants shouldn't affect the liver. some medications have more affect than others and the labels always state they shouldn't be taken by people with liver problems, this is because all forms of medication/drug are metabolised through the liver but as long as your dr is aware of this they can use this knowledge to aid them in choosing the correct meds at the correct dose.
Now that that is out of the way, could you possibly see your GP? I know a few people who have been to their GP and without a 'formal' diagnosis from a psych have gotten antidepressants and have never actually needed much, if any, intervention from the mental health services.
It's good that you are getting help, my advice would be not to see too many professionals as personally I have found this very stressful in the past as they always tell you different things (my opinion only!) If you can find one that you trust and know that is LGBT friendly I would stick with them.
As hard as it is try not to let your parents influence your decision, I don't know about you but even after several hospital admissions, 8 years of self harm and suicide attempts my parents don't know how I really feel (in fact they believe it to be experiments) so in my case my parents wouldn't be the best to advice... I am not in any way suggesting you don't include your parents but if you haven't until this point it is probably a shock to them.
I hope you can work something out which works for you :) and that what I have said makes a little sense and is even a little helpful!