Palms are not a good choice for a hammock as any suspension system is going to damage them. Plant a couple of trees instead on 15' centers - minimum. If you must use palms then I would suggest Royal Palms.

I was afraid of that.
We cut down 40-yr old laurel oaks, which sure makes pool maintenance easier for our neighbor, and a lot less stress in case we have a big blow, but now we have a big tiki hut in place.
Any suggestions on trees that fit a tropical landscape and don't drop a lot of stuff?

This is not the answer you were looking for, but why not just install two posts for the hammock, then plant a palm near each post to provide shade and the tropical look? You could arrange the posts and palms so that the palm trunks block you from seeing the posts, at least from one side. That way you could have your hammock and your palms this year.

You could camouflage or decorate the posts with garden art, or hanging baskets/half-baskets of tropical plants. You could also install plantings at the bases of the posts. Posts could also do double duty as supports for outdoor lighting - tiki torches or electric lights. (Another option is to grow vines to cover the posts, but you'd be forever cutting them back so they don't steal your hammock.)

Sabal palmettos are tough as nails. Nearly 99% of them were still standing after Hurricane Andrew blew through with up to 175 mph winds. Burlapped, hurricane cut ones will require 2-3 years to root properly with regular, frequent watering in sandy soil. The best ones are the ones that have been recently field dug. Plant them when temps. are consistently in the upper seventies or higher during the day.

L_FL : I think that is a great idea!! I may try that myself :)