I am sorry that this happened to you. Your record should currently accurately reflect that you were arrested and that the case was dropped. The problem is that when someone does a background check, they will either ask you to explain, or not even give you the opportunity to explain.
You do indeed have a right to have your record expunged, but you will have to pay your local police to have them prepare your fingerprint card for you, you will certainly have to pay FDLE their fee for the Certificate of Eligibility to have your record expunged, you'll have to pay whatever filing fee with your local clerk of court to file the paperwork.
Police generally make it a point to search vehicles with dark tint, because they feel that from their experience (profiling) people with dark tint are often doing something in their car that they don't want others to see.
FDLE's website has the forms and step-by-step instructions to do the expunction process yourself, but it would never hurt for you to talk to some of the great criminal defense attorneys in Tampa. The process is set up to make it possible for individuals to do this without an attorney, but many pepole do prefer to have an attorney do it for them for a fee.
Geoffrey R. Mason
I strongly agree that it's not right that you have to pay to clear your record of an erroneous arrest. Not to mention that you spent a night in jail. However, that's the law in Florida. Below is my guide to sealing/expunging your arrest- it sounds like you'll be eligible to expunge.
We need to get our legislature to change the law regarding record seal/expunge- but, until we do, the State is punishing people for their errors. It's an unfortunate situation I see all the time...