Brian,
The last time we went to St, Andrews (first week of June), we got off the train in Kirkaldy, called to get in the next days' daily lottery by 2pm and by the time we rented a car and arrived in St. Andrews, we had a time on the Old Course for 4:50pm the next day. When we arrived at the starters' shack to check our time, we celebrated our time for the next day, then asked the starter if there was availability on any course that afternoon. He looked at the first tee of The Old Course and said, "If you can be ready in 10 minutes, you can tee off there." We pulled our credit cards out and said, "Let's go." The Old Course that afternoon, then the New Course the following morning and our celebrated time at 4:50 the next day (w/o any booking agency). Hope you have the same luck we did.
Aloha,
Greg