Can I sue another party for the difference in appraisal of a total loss vehicle?

My car was rear ended in Florida and declared a total loss by the other party's insurer, which took 100% responsibility. I had an independent appraisal done, and that amount was $4,000 higher than their offer. I do not wish to go through my insurance, and so the other insurer will not negotiate. Can I accept and cash a check from them, sign over my title, and then sue the other party for the difference in small claims..... as long as I don't sign anything? Or is there some provision of the law that says accepting their check means I cannot sue for the difference? In Florida, apparently I cannot recover attorney fees unless I sue my own insurer, which seems absurd to me, thus limiting just suing the other party outright as legal fees would eat up any larger settlement. Thanks.