To add to what others have said- do be aware that the legal picture is different from country to country and (in the US) from state to state. The social picture is different too and may not always line up with the law! For example here in the UK bathroom use etc is a matter of self-identification according to law, but as a pre-everything MTF who doesn't yet pass I'm not about to go change in the women's locker room, much less use the communal showers.
I totally know what you mean about discomfort. My local yoga studio is a big part of maintaining my sanity, except for that few minutes before and after class when my brain is just a whirlwind of 'get me out of here', and I have a mental map of every unisex bathroom at my university. I would feel so much happier using the women's, but the last thing I want to do is make other women uncomfortable. (I don't care about making cis-men uncomfortable, lol.)
Workarounds for your workout: arrive in your gym clothes and sidestep the issue? If you think your gym is TG-friendly, perhaps explain your situation and ask if there's anywhere you can change privately?