Always accept the offer. If you don't, admissions will no longer be obliged to help you. If you accept it, you can join waiting lists for other local schools. There may even be other local schools with places currently.
Commuting 13 miles to school is madness so you really do need to be looking at getting them both into a more local school, rather than trying to get your daughter a place at the one 13 miles away. The chances of her getting a place from the waiting list are incredibly limited, given how far away you live. That commute doesn't sound at all practical for 7 more years anyway. It will eventually isolate your children, not being local to their friends. It might be fine when they're little, but not when they're in upper KS2 and wanting to walk themselves to school with friends and socialise with them out of school.
I understand it's distressing to see your child unsettled and upset at a new school, but you need to give them time to settle. Moving them after a week, if they do struggle to settle, is really confusing and distressing for children. Sadly, when it comes to this age group, there will be odd incidents of children hurting each other, whether by accident or on purpose. Has your child never been hurt at his other school? Children need to learn to deal with conflict. Obviously if incidents keep occurring with the same child that is unacceptable, but you work with the school to raise the issue and ask for their support in dealing with it. Moving your child every time they are hurt would be crazy.
I get that it might be awkward to try to get your eldest back into the local school your daughter has been allocated. However, you have time now to visit other local schools with spaces for both of them. Hopefully you find something you like. Then you approach the move with the utmost positivity and give them both time to settle in properly, accepting that a bit of distress on your son's part is normal and expected in the circumstances. Change isn't always easy and seamless. However, he can learn to be happy elsewhere, with new friends who live in your new area. Sometimes it takes a child a whole term or more to accept their new norm and feel happy and settled in a new school, especially if they're very attached to the old one. That doesn't mean they'll never be happy. You just need to give it time
Good luck