First, let's explain what you're referring to when you mention a "dram shop" law. These are state statutes that can be used to hold a business liable for providing alcohol to someone whose intoxication goes on to play a role in causing some kind of accident (such as a drunk driving accident). Not every state has this kind of law on the books, and the states that have passed them come down pretty differently when it comes to the details of liability.
Florida's dram shop law can be found at Florida Statutes section 768.125, and it's pretty limited in terms of when a business (whether we're talking about a restaurant, a bar, a liquor store, or some other establishment) can be held liable for damages when an intoxicated customer goes on to cause an accident. Section 768.125 actually explicitly prohibits this kind of civil liability, as long as the customer was of legal drinking age, except where:
How does this law work in the real world? Let's say Paul stops at Danny's Liquor & Spirits to buy a six-pack of beer. Paul is 25, Danny checks his ID, and the sale is completed. Paul drinks most of the beer right in the parking lot. A few minutes later he runs a red light and collides with Vicky's car, injuring her. Police come to the scene and administer a Breathalyzer to Paul, who blows a .19, meaning he was driving under the influence of alcohol.
Vicky can definitely file a lawsuit against Paul for causing the car accident. (Learn more: DUI and Fault for a Car Accident.) But can she also sue Dan's Liquor & Spirits? Not under this set of facts. Remember, Florida's dram shop law specifically prohibits this kind of civil liability on the part of alcohol vendors, without evidence of one of the two exceptions highlighted above.
But if we change the facts so that Paul is 17 and Danny doesn't check Paul's ID even though he knows Paul still a high school student, then Danny's potential liability also changes. This would almost certainly qualify as a situation where a business or its employee "willfully and unlawfully" sells or furnishes alcohol to someone who is not of legal drinking age. That means Vicky could most likely file a civil lawsuit against Danny (in addition to the case she can bring against Paul himself) asking for compensation for her injuries, vehicle damage, and other losses stemming from the car accident.
Under this revised set of facts (where Paul is 17 and Danny doesn't card him), if Paul is injured in the accident, he may also be able to rely on section 768.125 to file a civil lawsuit against Danny.