You haven't allowed this to happen.
It really isn't you, it's him.
You are getting out, you are getting a safe place for you and your child - that's you.
The council have private rooms and you will be able to request to speak to female if you choose to. There isn't anything they haven't seen and dealt with before so don't worry.
Are you on medication for your anxiety, or are there any breathing techniques etc you can do that help a little (I really don't know too much about anxiety so maybe someone else can help with some tips)
You have planned the hell out of this already. He is away, you have hours before he realises you have gone, you have your journey planned, you know where the council is, you have things sorted for your child, you have even planned the shop you ate going to for credit.
All that's left to do is follow your plan.
When I left I felt conflicted for ages afterwards. I tried to tell myself it wasn't so bad and I'd lived it for nearly 15 years so it can't have been awful etc. Now, looking back, it was actually worse than I imagined. I did the freedom programme which helped me see a lot too.
Don't think about your whole future tomorrow. Just keep thinking about the next step, then the next one, a little bit at a time 