If I move out-of-state before my divorce is final, will it hurt me in court?

Not until an agreement is signed.

I'd have the agreement reviewed by a lawyer, in Florida, to be sure it's properly executed and otherwise enforceable.  Assuming it is enforceable, then moving out before the divorce is filed won't change that.  The real question is a much more practical one:  can you trust your husband to honor the agreement, about the rest of the items, later on?  Having to go to court to force him to do that would be a real headache, and maybe not even worth it, depending on what the left-behind items might be.  Of course, you can do an end run on that, too, by renting a storage unit, and removing what's yours before you go to Ohio.

If your agreement was properly drawn up, it should include a provision that specifically gives each of you the right to live separate and apart.

Courts like settlement agreements.  They like them a lot, especially in divorce cases, and usually will enforce them if they are reasonable and fair.