To directly answer your questions: yes, Gulf of Mexico
(GOM) crossings by cruising sail boats are common as is them looking for crew. So it is feasible. Post in the Crew Wanted
Consider seasonal timing too: not many want to brave a GOM crossing in hurricane
months can get nasty too (...and the GOM can get epically nasty)...the earlier you can jump in winter
months the better to avoid frequent and strong fronts, but Nov-Dec is not uncommon to get down to, or back from, the W Carib right after hurricane
season. The spring season is the most settled. Now thru early June is a good time.
Also, in my experience most head for Isla from the Texas
coasts...you might fare better looking for a ride from there. You could bike/hike/paddle to either from the Mississippi
Of course, you can also go self propelled thru Mexico or along the Mexican coast (long way round though).