HHI is an awesome place to play golf. All the advice you have here is on point from a golfing perspective. I've played all the courses people are mentioning and will give you this input:
Hilton Head Nation- Its not as sexy and prestigious as Harbor Town, but the people there treat you awesome. And as a value, last time I played it was about $60. There are no homes around it, they have 3 nine hole layouts, and I personally enjoy Gary Player tracks. I say it's a must play.
Palmetto Dunes- The HH ads always feature a green on the ocean and that shot is from the Palmetto Dunes course. There are 2 courses in there, the Dunes and the Arthur Hill. I played the dunes and found it enjoyable, but the only ocean views are on that one single hole (cant remember the number). It's a fun track, but its a typical resort layout. Also Palmetto is mid island so you can get to it easily from end to end depending on where you are staying.
Shipyard- In the trees and with lots of homes/condos all around.
Harbor Town- I will call this a so/so option. It's very expensive and the only views of the waterways is on 17 green and 18. The rest of the course is cut in the trees and condo land. For me, I was disappointed. Yeah the tour plays there. But there isn't anything remarkable about it except 18 and the price. If you want to play the "name" courses, then you should play here. But you can play HH National for about 4 rounds for the cost of one round at HTGL.
Tyler, HHI is one of my favorite places in the USA. I plan to retire down there. It's not as uppity as Kiawah and not as touristy as Myrtle Beach. It's a really great area and once you know your way around it's awesome. Tons of awesome golf and things to do. Have a great time.