Scott:
My experience is from 10 years ago, so it comes with a mild caveat that things may have changed since then. But, I was there during the week right before the Open in July, held at Carnoustie, when many players stayed in St. Andrews and took the hovercraft across the bay to Carnoustie. In other words, the place was crawling with golfers and golf fans.
There is usually a long line of singles who get up quite early and wait to give their name to the starter at TOC. I waited this out, had breakfast, and instead wandered down to the starter at @ 9:30 am -- on a perfectly sunny day, 70 degree temps, no sign of rain, perfect golfing weather. He asked me to come back at @ 1pm, and he would get me on within 20 minutes or so.
I think it's very likely you could get on as a single. Another tact might be to hang out the night before at one of the nearby watering holes -- the Dunvegan is notable for the golf crowd it attracts -- and talk up anyone there (maybe even a caddie) to see if they have an opening, or what may be available.