If you choose the Palmetto Dunes courses, make sure to first select the Robert Trent Jones course. I double checked my notes and that is the better of the two. It's also the hole pictured for many Hilton Head ads, where golfers are on the green and the ocean is right there.
That is the signature hole, a lovely par 5, where your approach shot is straight at the ocean. When I went to HHI we always stayed at a home along the beach in Palmetto so we could walk to that hole and sit and watch.
As for HH National, it is a hidden gem. Everyone always talks about Sea Pines and how great it is. It is very narrow and "condo golf" for the most part. The beauty only comes on the last couple holes when you're out on the Sound. HH National has more natural beauty, as it's cut out of the forests, and there is nothing around it at all. Best of all, it's under $80 to play, so you can play it 3 times and not equal one round at Sea Pines.