Buck W.,
It is hard to argue with what Stan and Brent have suggested for a week in Dornoch. Playing Royal Dornoch (green fee about 70 pounds) 7 days in a row could hardly be viewed as a bad thing!
However, what I find equally as attractive to the area are the other local courses (greens fees in the 25-30 pound range). Brora, Golspie, Tain and the improved Struie Course at Dornoch are all worth playing, as is Fortrose & Rosemarkie, another James Braid course about 35 miles south of Dornoch.
My suggestion would be to play Royal Dornoch at least 4 times, buying a day ticket that gets you on the Struie Course at a reduced rate at least two of those times. Then I would play Brora, Golspie and Tain once each and return to whichever of those courses you like for a 2nd or 3rd round.
Personally, I like Golspie, as it is in a beautiful spot and the course does not seem to get nearly as much play as the Dornoch courses or Brora. If you show up before 8:00am on a weekday morning, the odds are good you will have the place pretty much to yourself.
The one time I played Nairn, I was not all that much impressed with it relative to Royal Dornoch. It is a 75-minute drive from Dornoch to Nairn and I would just as soon spend those 2+ hours playing an extra round in the Dornoch area.
But that's just me!
DT