OP I could have written your post word for word as I have a cousin that is like this. We were really close once but as we got older the neediness and attention seeking got worse.
She used to come round for hours on end, every conversation was turned back to her, whether it be relationships/health/money/work etc it would all be "Woe is me, how shit her life was. She also used to bad mouth everyone and anyone (usually down to jealousy) which wasn't nice to hear. If she wasn't getting the attention she craved she started making up serious issues/incidents to gain that attention, then would drop herself in it by contradicting herself. It got to the stage where I no longer knew what the truth was. (i'm not even sure "she" knew what the truth was tbh!!!)
I was left feeling absolutely wretched after ever visit.
Two years ago it all came to a head when I was going through a really shit time of my own and although she asked how I was, didn't really care for the answer. She had come round and spent the best part of 4 hours telling me her (non existent) problems. When she left I broke down in tears to dh. I couldn't do it anymore, I had nothing left to give and clearly remember saying to dh "I don't know what she wants from me" She'd finally done it she had broken me.
I started to withdraw, stopped being so available. I needed time and space to fix my own head (I too suffer from depression but can usually deal with it without going back on my meds, but I need time out from other people to do it).
Not long after, Dh had a word with her and told her a few home truths when she "demanded" to know why she couldn't come over.
It took me months to get back to a happy place again and I needed to understand why she was the way she was so after a lot of googling I finally think I found what I was looking for. That made me realise that it didn't matter what I did or didn't say she was never going to change .
Now when we spend time together it is on my terms, as soon as she starts with her "woe is me" crap. I just smile and nod and then change the subject or if it is a problem of her own doing, I tell her so. If I suspect she is lying and she contradicts herself I point it out and thus it goes. I don't play into her hands
I cant go no contact but It helps me massively to know that I can deal with her and she cannot take from me something I am not willing to give.
I think she has finally realised this and its not all doom and gloom now which is much better.
But having said that if she were someone id only known a few months I would have phased her out slowly and permanently a long time ago.
Its really not worth your own wellbeing "trying" to be there for someone who gives nothing back so don't for one minute feel guilty OP
Put yourself first!!