I have done a video on this in the past about gym Anxiety.
But I want to come at it from a different angle.
There will always be something stopping you from starting and taking that first step.

Be it getting a gym membership and using it.
Contacting a Personal Trainer.
Signing up with the Personal Trainer you have contacted.

There will always be a nagging voice in the back of your head stopping you from ever starting or starting up again.
As soon as you start up you will feel amazing.

It’s just that first step. As soon as you have taken that step you will be grand.

My Advice would be this. And by no means am I’m an expert, but to me this is logical and it should help.

Environment.
Find an environment you enjoy being in and around. For example, the gym I work out of, Is full of great staff, it’s light and airy, There are no egos or ‘Roid Heads’, Its fairly quiet with no crazy busy times (As random as that sounds!).
Your environment will dictate your success.

Timings.
Aim to try and find the time which makes you feel most comfortable. Be it Early, Midday or Evening. Aim to find one which best suits you and makes you feel most at ease. This could be because there will be fewer people, or early because everyone is still half asleep at the gym.
Experiment with it, but don’t rule one time out on one experience as this could be a fluke. Try each time 2-3 times and then judge it.

For ones who have signed up to PT.
From my experience, someone people get nervous about what their coach has planned or feel like their coach will kill them.
We don’t kill people, that involves too much paperwork.
I can only speak for myself, but I pitch my sessions at slightly above the level the client is at. This means the client can do it, but they are slightly pushed. So you will not need to worry about that.
If you are still worried about the sessions, you can always text your coach and say ‘Hey, Just seeing what the plan is for today’s session’. Personally, I would have no issues with that at all.
If you do not feel comfortable with the coach then say so and see if the coach can adapt and give them a chance to adapt as well but if it doesn’t work out then say so to the coach. They won’t hold it against you. End of the day we can’t click with everyone.