Despite the thaw in relations between the west, in general, and the Republic of Cuba travel onward by private sailing boat is still a rarity. There is a race between the US and Havana in January and I think there will be some sailboats entering the race which will continue beyond Cuba rather than returning to the US. Cancun in Mexico would be a logical trip as it is only about 300 miles. There may also be some boats currently in the Caribbean which may be heading for Cuba on their way toward the Panama Canal.
I think taking on new crew in Cuba could still present some bureaucratic problems when sailing into the next foreign port. US flagged vessels still are closely scrutinised and regulated when it comes to Cuba.
Sponsoring a lift to Cancun from Cuba could be very difficult for owner/skippers to countenance due to existing perceptions and continuing restrictions. The winter weather for such a short trip could make for a nice trip with northerlies at 10 to 15 knots currently blowing.
I wish you well and hope you can get a ride, but I think you may be better served, given the time frame you propose, in paying for a cruise from Havan on board one of the cruise ships which regularly ply those waters.
"if at first you don't succeed....Redefine success"!