It sounds like the clerk made a mistake and did not follow the Court's order at the time of your plea and sentencing. You can retain a lawyer to file a motion to correct sentence. A hearing will be held and you can show the court minutes from the original plea date which shows the careless driving should have been dismissed. The State Attorney will also be present at the hearing and will likely agree that it was the intent of the parties to drop the careless ticket if this was a plea. The Judge at that time can make a more specific order explaining the merger and dismissal in order to correct the record. The Clerk can be ordered to forward the court's order (dismissing the careless driving ticket) to the DHSMV and they should correct your record at that time. If not corrected, the careless driving ticket will remain on your record in Florida for 75 years.
The above attorney is correct. You most likely need to file a motion to correct sentence and let the court know the situation. Also it would be a good idea to pull the audio record of those proceedings to see if there was a mention of the agreement on the record.