to hear your story!
I believe that you can apply to any state school you like, and appeal against the decision if your application is turned down. The main problem with admissions for this age group is presented by the the regulations on infant class size, which impose a maximum number of pupils in each class and might make it difficult to find places in another state school especially as you have two to think about.
Nevertheless, if you are turned down and decide to appeal you would have the opportunity to put your case to an independent panel who would consider the circumstances and come to a decision which is binding on both sides. The general rule of thumb for school appeals seems to be to explain why your child needs to attend that particular school and not to offer negative comments about the school they have been offered or are currently attending... though in your case that would be difficult! You would however have some good points of reference, e.g. you are able to show why you would prefer a larger school. I know that appeals are sometimes upheld in cases where a child is moved due to bullying issues etc.
The independent school option is another possibility as there is every chance that you might qualify for a bursary. It is certainly worth looking into.
Good luck! Your children really should not be having to put up with this kind of thing.