Since you are arden golfers, I wouldn't give up on the 'Old' course, quite yet. Yes, I have heard you need to book long in advance, but we were there in October, a few years ago and DH decide to walk up to the little green structure where you register and ask. Chance is a fine thing..A trio was just about to set out, and after asking, yes, they welcomed him..He hurried and rented clubs, etc, and had a wonderful time..I went off to have the best cup of coffee that I had in Scotland at a nearby tiny place (TASTE) on North St.. cross street, Murray Park, and then a few hours of happy shopping and wandering in Saint Andrews. Yes, it was softly drizzling for much of the time, but that was fine. One of DH's highlights of the trip, and a framed photo. Maybe the luck came because it was October, and maybe May would be different, but doesn't hurt to ask.

"Yes, I have heard you need to book long in advance, but we were there in October, a few years ago and DH decide to walk up to the little green structure where you register and ask. Chance is a fine thing..A trio was just about to set out, and after asking, yes, they welcomed him. . . . Maybe the luck came because it was October, and maybe May would be different, but doesn't hurt to ask."

That is the way 95+% of singles play the Old Course. The starter (the man staffing that 'little green shed') will hook single walk-up golfers with the next available two-some / three-some. It's doable pretty much year round.

But sadly I'm pretty sure the OP's trip is off :( since it was scheduled for May this year. Ain't no one traveling to the UK in at least the next two or three months if not much longer.