I'm off (to China) for a couple of days. Back on Thursday. I'll post the answers then. Some of the above suggestions are right, some wrong.
A few more clues:
One of the clubs started in a coffee house.
One of the holes has been (on and off) the longest in Open Championship golf.
One of the courses was (we think) designed by the club secretary.
The oldest and second oldest links clubs in England are both there.
One of the courses is attached to a hotel which was formerly the local maternity hospital.
Where golfers now tread the fairways of one of the courses formerly potatoes were cultivated.
Parts of one of the courses are below sea level.
Tony Jacklin won his last European tour event on one of these courses.
One of the courses was formerly a royal hunting ground.
One of the courses had a 70 year gap between American winners of the Open Championship over its links.
One of the courses is one of only three to have hosted the Ryder Cup, Walker Cup and Curtis Cup.
One of the courses was the first to charge an entrance fee to watch a professional match.
One of the courses was particularly influential in early American golf design.
Two of the courses used to have a railway running alongside, three still do have.
One of the courses is in a Cathedral town, another has the name of a Cathedral city and a third has a ruined cathedral.