I really feel for you OP as I have struggled with social anxiety (any anxiety!), stress and depression for years and I can see how easy it is to think HE would be the solution to all your problems. However, did I read that Dad takes them to museums etc on a weekend but you don't go as it is too overwhelming? If so I'd be worried that at some point , going to the HE centre is going to become too overwhelming too. Apologies if I read it wrong.
What I can really see happening here is you isolating yourself further and far from being a solution to your problem, you will be making yourself worse. Whilst the children are so young, this may not seem so bad, but once they are older they may want to join clubs, make more friends, have other experiences that are further from your doorstep.
I get overwhelmed and anxious easily if I don't feel prepared for the morning. I'm lucky enough to work now, but I have to ensure packed lunch is ready, uniform is ready, bus fare ready if I need it, breakfast things ready - all the things that need sorting of a morning, but the night before otherwise I do get very stressed.
We now walk 1.5 miles to school and I walk 3 to work, and the same in reverse - I cannot tell you how much this has helped me mentally.
If you feel you cannot do this, could you look into paying somebody to do the school runs for you? Are there any other local parents? It's taken me 3 years of thinking no other family lives near me, to meet a mum on the school run, become good friends and share the burden between us.
I'm sorry but I agree with the others, this seems like a solution to your issue, in which case I thoroughly recommend looking into some CBT or counselling. Yes of course your anxiety will decrease if you don't have to be somewhere for 8.30/3.30 on the dot involving 4 buses a day BUT I guarantee that nothing will get resolved if you use this to isolate yourself further.
Wishing you all the best x