Hi Dayle,
Congrats! I'm glad to hear HRT is working well for you. I'm envious....I haven't been able to start my program yet.
When I present as female or feminine androgynous, I always use women's facilities or a gender neutral bathroom. I go in, do my business, and leave. If I'm nicely dressed, I sometimes get compliments and will acknowledge them. I've never encountered an uncomfortable situation in the women's facilities. These days most women won't make a fuss if they see someone who's trans. If it happens, just get out quickly.
Regarding your gym, talking to a female staff member might help to ease your apprehension. Most likely she'll tell you it's totally cool to use the women's locker room. If you get a negative vibe, you should probably start looking for a new gym.
I strongly believe it's unsafe for transwomen and androgynous males to use the men's facilities. There are 2 clear dangers....running into a transphobic male....or running into a misogynistic male. Either can quickly ruin your day!
Hope this helps.
Hugs, Emily