Sorry, just saw this message.
You'll want to go play the May River GC at Palmetto Bluff
http://www.palmettobluffresort.com/golf/overview/
It's about 40 minutes from Hilton Head, but I can assure you the Nicklaus course is well worth the drive. And, yes, it's open to the public. No idea how much you want to spend, but it is one of the best public courses in the country.
Also, I'd recommend looking into playing either the RTJ or Fazio course at Palmetto Dunes.
To be honest, I'd just bite the bullet and drive 2 1/2 hours and make a day trip out of going to Kiawah Island. Not sure where you live, but playing the Ocean Course at Kiawah is a once-in-a-lifetime experience. But that's just my opinion...