Florida has a similar statute to that referenced by my Georgia colleague--check out Section 701.04(1) Florida Statutes. That statute gives the judgment holder 60 days to record what in Florida is called a Satisfaction of Judgment and to send the recorded SJ to you. If the creditor does not do so, you can bring suit and have your attorney's fees assessed against the creditor, so your writing a letter to "remind" the creditor that a Satisfaction is still due is a great idea. Just be sure the Satisfaction has not already been recorded, but simply not sent to you--makes a big difference overall.
Here is some bad news: the recording of the Satisfaction may not have an effect on your credit score. The judgment will remain on your credit report for 10 years from when it was entered, even though it is been paid off.
Why? The fact that a creditor has sued you is considered to be a sign of poor judgment and/or exercise of financial discretion. You can not force the credit bureau to remove the judgment from your record.
Regardless, you would want the public records where the judgment was recorded to reflect that it had been paid off.
Good luck in your new home.