What should I do to protect myself in case he changes his mind after agreeing to me moving with the kids?

You should have an attorney prepare an agreement memorializing his consent to your relocation to the city and state you're moving to, and setting forth what the new timesharing arrangement is going to be, as well as any other terms and provisions that are necessary under your circumstances. Florida has a relocation statute that is strictly applied in these situations, so you really should have an attorney do this. Consult an attorney to discuss the specifics of your situation and your options.

Get his permission in writing to do this. Just agreeing is not enough, it either has to be his written permission to move out of state or a court order. But then I know only what you have written here and it does not indicate whether you are or were married to him, which can make a big difference. The best thing to do is to have a full discussion with a family lawyer.