I went full-time at three months on HRT, and didn't get my legal name change until another three months after that. I am still working on getting all my accounts changed.
I am surprised that you had a problem presenting ID. No one ever asked for my ID. I pay for stuff with my bank card, which I insert into the machine myself. No one else even sees the card.
I did have one amusing incident at the bank before my legal name change. For whatever reason, I had to talk to a real, live, human teller. I was full-time and presenting as a well-dressed middle-aged woman. I gave her my card, complete with male name, and she swiped it and started reading my record on the computer screen. She stopped, with a puzzled look on her face, and said, "This card you gave me, it isn't yours!" Oh, crap, I thought, here we go. I told her that it was indeed my card and that I just hadn't had a chance to do my legal name change yet. She said, "No, it's not that. There's no way you are 62!" With a big smile on my face, because she had just made my day, I thanked her and admitted that I was indeed 62. No issues with the name, and she even offered to record a preferred name on my record.
I did get my name change document, and I have updated my driver's license (and bank card). It shows me as Kathleen Lauren Xxxx, Sex: M. Argh! Darned British birth certificate! So, if I get stopped by the cops, I might have some 'spaining to do. But the Mounties are very polite, and there's a good chance that the combination of my feminine presentation and the obviously female name Kathleen will clue them in to use feminine forms of address.
I won't tell you what you should do. But I would recommend that you not delay going full-time just because the paperwork is slow.