Oil rigs will be a-penty, but relatively easy to see. Once you are beyond about 30 miles you will be free of them. You are likely to be free of any wind though also, and it gets very different.
We found a big problem in the Gulf of Mexico, in July-August. If you try to cross in a straight line to Florida Keys, the entire distance was, for us, one vast, eerie, calm. Every night the thunderstorms arrive, and they were legion, marching at us in strange lines. Watch the thunderheads with the low, pancake-like clouds beneath them... they start the sparks at about 5 pm, and continue until just before dawn.
If I had been alone, I would have been chewing the anchor rode. Flapping sails and dancing blocks rattling on the deck drove me half insane.
The motor ran for nearly 90 hours, if I remember. There really isn't much wind at all, and what very little we had was from 120 deg, and right on the nose.
If I did not have a reliable motor, I would coastal-hop, though no-doubt that will lead to different risks.
You will get about 1 kt assistance from the Gulf Stream, and it's welcome.
It's a long way to Dry Tortugas, about 650 miles, but it is a paradise....
Image of Dry Tortugas, Summer 1992 - Photobucket - Video and Image Hosting
...that's my ship there, in 1992.
Pls be careful, especially alone. If the wind drops and your motor plays up, you may be stuck out there when the weather turns nasty when one of those tropical horrors stirs things up.