Tim -
One of the greatest weeks of my life was a week spent almost 22 years ago, traveling by myself & playing golf around Ireland. As I recall, I don't think I booked any tee-times in advance. I simply showed up (at many of the courses you intend to play) and asked if I could play. There was rarely a problem. But that was well before Ireland was "discovered" as a golfing destination.
One thing to be aware of is that many clubs in GB&I reserve the 1st tee for their members from 9am to 10am and then from 11am to 1pm. In most cases, they will not let visitors tee off in those times, even if the 1st tee has been empty for 30 minutes or more.
In addition, tee-times can frequently been taken by the members for league matches or inter-/intra-club tournaments and events.
As noted above, your best bet as a single is to get there early. In my experience, if you get to a golf course in GB&I by 7:30am, you will be the 1st or 2nd car in the parking lot most mornings. At St. Endoc in England, I paid my green fee in the afternoon to tee off at 7:30am the next morning, because the pro shop did not even open until 8:00am.
Finally, some clubs (Royal Dornoch, for one) only permit 1- or 2-ball golf games before 9:00am most mornings. This is done to keep the pace of play moving along at the start of the day.
Have a great trip(s)!
DT