Yes, it's the score that counts, 9 hole scores being doubled (e.g. 33 => 66). Because only a limited no. of the best scores will be included in the average, it's also important which old scores may be replaced. This determines how much the ave. drops and if it will drop at all.
Some may have noticed that a score above their average caused it to drop unexpectedly. This is for example, if there is a score of 80 in an average of 72, a score of 76 (or 79) will kick the 80 out and thus push the average below 72.
A look into Stevie's score history reveals that the highest scores in his average are 66s (15x) and 64s (19x). Thus, now scores of 65 (above the average!) and better will move the average until all the 66s are pushed aside. Eventually, only scores below 64 (that is the 32s you mentioned) will help.
Best to drop the average? You mean "fastest"?
I heard that quite a few players play only StAndrews front nine for that purpose, others say Kiawah is easy. A lower course Par helps if the course ain't too difficult.
I once found a good motivation in chasing the course awards (18 holes), but 9 holes may be easier. Playing the multi-play weekly and monthly tournaments pushes the XPs as a side effect.