First off - theres no 1 definition of gender dysphoria, but I've experienced what you've described and I've been diagnosed with gender dysphoria by several psychs (I am non-gendered).
2nd - facial hair; the cheapest quick fix is make-up (concealer, foundation) but this didn't work for me so I went for laser removal and now electrolysis to get rid of whats left. The cost of laser can be reduced if you block-book several sessions, it also reduces as the amount of hair reduces - electrolysis is usually charged per minute, or for 10/15/20/etc minute blocks. TBH it does cost a fair bit overall but the cost is spread out over time so its really not that bad, note that laser appointments have to be 6 weeks apart. The overall cost of my facial hair removal has been about £900 so far for 10 laser sessions and an hour of electro, but this has been spread out over about 2 years and there is very little left (BARELY visible a couple of weeks after shaving and I shave maybe twice a month now, used to be every day and still had a shadow). My quality of life has improved so much as a result, I wish I had done it years ago instead of moping about it. Body hair naturally reduces quite a lot with HRT, I thought I would have to spend 1000's on laser for this but I haven't needed any (been on HRT for 2 years).
3rd - identifying as female; this is something that used to concern me too. Personally I tend to focus on taking care of the physical dysphoria and not worry too much about whether I "am" female or male or whatever; as a result of HRT and hair removal I am now perceived as both male and female but I don't feel like either, I tend to go with the flow and if the subject of my gender/sex comes up I say I'm non-gendered. Maybe you should try presenting as a woman and see how you feel about that and about how people respond to you? I did that pre-transition but it didn't work for me because in order to be perceived as a woman I had to do stereotypical things like wear make-up and skirts and I way prefer jeans and t-shirts (which is what I wear now and I am sometimes perceived to be a woman anyway so...)