Firstly, congrats on getting your name change through! Woohoo!

As you know, I've been using the gents' exclusively since last year... and you know what? I still get nervous every time I have to go in there. But so far, after almost a year of using the gents' - most of it pre-T - the worst thing I've had happen to me is that I've had to wipe a bit of mess off the seat before I could do my business.
I've never been confronted.
I've never been challenged.
I've never been laughed at.
But in spite of this, I still get nervous. I'm only human. And I spent 40 years using the ladies' room, where women stare at each other intently and take note of every little detail of what you're wearing, doing, saying etc.
Guys don't give two hoots about any of that stuff. Those people who stared at you were only staring because you were clearly distressed. It's only natural that you should feel that way; it takes time to get your head around the fact that you have every right to be in the men's room. You're worried that the guys are going to check you out (like the girls do) and that they'll notice that something isn't quite right. But the truth is: guys don't even look at each other in the gents'. It's almost as if they're afraid to look at each other, in case someone might think they're gay. So they keep themselves to themselves, do their business, and get the heck out of there as quickly as possible. Which, of course, is what you want to do too.
To help you cope with your nerves, it's a good idea to move gradually from the ladies' to the gents'. That's what I did, and it helped my nerves enormously.
When I decided to stop using the ladies', I started by using only gender-neutral toilets. I know a lot of the Disabled facilities are locked (and I know that it's difficult to get the Radar key to open them), but quite a few retailers have gender-neutral facilities. Costa Coffee tends to have them, as do most branches of Subway. I recommend doing a recce of the local shopping centres etc. to see whether they have family rooms or unlocked disableds so you know where to go (as a suggestion, that big, new shopping centre we've talked about has family rooms and disabled rooms in the ground-floor toilets, and you don't need a key to get into any of them).
Then you can graduate onto quieter, less used (or single-stall) men's toilets. A good suggestion would be Starbucks, which has single-stall gendered facilities. Many of them have single-stall men's rooms in a discreet location; I'd recommend giving them a bash. This will help you get used to going into the men's room without having to share the space with other guys.
Then, move onto quieter multi-stall men's rooms. Most shopping centres have a set of toilets that are in an inconvenient location so they tend to have fewer people using them. That big, new shopping centre has a very quiet set of toilets on the top floor in the management suite; try using those when you're ready to graduate onto multi-stall facilities. It's very quiet but there will be one or two other guys there, so it's good practice for when you're ready to start sharing.
Once you've done all that, you can move on to the busier loos - but there's no rush so just take your time. Several people have suggested just going in there to wash your hands so that you can get used to being in the room. That helps too.
I have some coping mechanisms that I use whilst in the men's room to help me feel better, like lifting the seat up after I've had a number 1 (so that it looks like I stood to pee) or staying just a bit longer, and going for the toilet paper several times, to make it sound like I had a number 2. These little confidence tricks help me get through my nerves. I also tend to adopt an attitude when I go in, acting like I own the place (without being obnoxious). This helps prevent other people from questioning me, no matter how nervous I might feel inside.
Above all else, take it
slowly and go at your own pace. Don't beat yourself up about the fact that you get nervous, and if you feel a panic attack coming on, perhaps it's a good idea to walk away for a couple of minutes to calm down, then try again at a quieter location. But please keep pushing ahead - gently - because using the gents' is the way forward for you.