Patrick
One of East Lothian's greatest assets in the number of quality links courses it has within a relatively small area. I'm a bit biased, as I work closely with the golf industry across the region but with 22 courses, mainly along a 30 mile coastline, there are many great golf experiences to be had. As the best links are located so close to each other, you can cram in a lot of quality golf with little time wasted driving from course to course.
I spoke to Anne McCarthy at Muirfield on Friday and they still have tee times available throughout April on their visitor days, which are Tuesdays and Thursdays. You can play fourballs from the 10th tee or 2 ball foursomes from the 1st tee in the mornings. I think the best experience is to play fourballs from the 10th in the morning and alternate shot in the afternoon with lunch between rounds.
North Berwick and Gullane No.1 are the "must plays" when visiting the region but there are several additional Open Championship Final Qualifiers definitely worth playing if time permits.
Dunbar is a classic Old Tom Morris design, which has 14 holes along a narrow strip between a wall and the sea. Luffness New, also an Old Tom design, where they have carried out some recent alterations, removing buckthorn and add a little bit length to a few holes. Gullane No.2 is another qualifying course and the club offers a discounted green fee when playing No.1 and No.2 on the same day.
You might want to consider staying along the stretch between Aberlady and North Berwick rather than Edinburgh, as this would bring you within 5 to 20 minutes of the majority of courses in the area. Traffic in Edinburgh is pretty chaotic at the moment, as construction work for a new tram system is underway. If you want to visit the city, it would be better to take the short train ride from North Berwick or Longniddry.
St Andrews and Carnoustie are both about 2 hours from East Lothian.
I'd be very happy to pass on more local knowledge and help with your trip planning if needed.