If your Nan is a well, strong woman then tell her everything. Tell your sister too and others you trust.
I wouldn't focus on moving, it takes too long. Just send him packing. Make sure you're getting all the benefits you're entitled to. Can you look at extending the length of your mortgage?
and get a lodger. DD sleeps with you & you'll just need to not dive into trusting your lodger to babysit etc.
look at possible childcare options so you can do more hours at work.
Record him if you can, but just get him gone.
Hes done a number in you, more people will believe you than won't.
when he's gone I'd write down everything you can remember from when you got pregnant until now, I wouldn't want him getting unsupervised contact.
get him out, today. Apart from all the mummy nonsense, he's been violent to you & that will ramp up.