All of the good diagonal routes, whether across the northern Midwest and starting down through Chicago, or going through Salt Lake City, through Colorado, Kansas, Oklahoma, Arkansas to reach the Gulf Coast in Louisiana or Mississippi, are about 3500-3600 miles, 50-55 hours driving time. Because I don't drive in winter conditions after dark (too difficult to assess road surface conditions) that would be 5-6 days each way for me. You will be in winter driving conditions shortly after leaving the Pacific Coast until you reach the Gulf Coast. Major highways are kept clear, except during actual storms when the crews are trying to catch up. You may have to wait out those situations, sometimes 2-3 days.
You can avoid most, but not all, of the winter driving going down along the coast to southern California and across I-10, which will be subject to freezing winter weather only between Arizona and Louisiana. This route is much longer, maybe 70+ hours drving time. It is popular for folks making seasonal moves, with time for leisurely travel and schedule flexibility, but does have less risk of delay.
Drug stops? I've no experience with that, but any travel route within 100 miles of a U.S. International border is fair game for immigration checks, whether random stops or checkpoints where all traffic is stopped for inspection. I've had to go through inspection at the permanent (but not always open) Homeland Security checkpoint on I-10 east of El Paso. I've also had to stop for agricultural inspection, several states do this, including Florida, but timing is random.
If this is a drug run for you, I suggest running through the middle of the country, and keeping your nose clean on traffic behavior, rather than trying to get across the areas where checkpoints are common.
I travel by road in winter from the middle of the country to the Great Lakes, to the Gulf Coast, and to Florida. In the winter I can manage 300-400 miles a day, adjusting routes and speeds to weather. Summer trips to same places, with 14-16 hours of daylight and only thunderstorms to dodge, I make 600-800 miles a day.
Your trip sounds like 5-6 travel days each way in the best of conditions, unless you have enough drivers to do it non-stop, the 3-4 days.
And if you want to see much of anything in Florida, consider that a trip from Central Florida to Key West and back was a week long for my wife and I, going down the Atlantic Coast from Cocoa and coming back through the Everglades and south central Florida. Things move slowly in Florida.