Wig wearer here! Yeah, it sucks like a super massive black hole, but I wouldn't pass without it so I just see it as part of my wardrobe. Like the others say, wigs are primarily made for cis women so wearing one doesn't make you fake.
During my first go at transition I was in a similar state, crying all the time, feeling like a phoney, after two years feeling the HRT had done a big fat nothing (it had, I just couldn't see it), and so on. A fair bit of the emotional stuff, I now believe, came from my HRT and delivery method (injection); I say that because with this transition I theoretically have more to be upset about than before (no hair, for starters, older, etc) and yet I've never felt more stable. So yeah, no injections this time (pellet under the skin instead) but I also have a better support network - counsellors, shrinks, friends, family, this forum - and I make good use of them, but I'm also on an anti-depressant which actually doesn't make me feel like a zombie but stops me spiralling down to the depths I reached before.
You've only been on HRT for a relatively short time, not really much to make a huge difference to the look of your face or body. Please, you need to give it time. The same for beard removal - god, mine is still months away from being dispatched and I've already been at it for months. I know that doesn't make the here and now any easier to cope with, but you will get there.
Hugs.