The shuttles didn't see that much use, and if they did you can bet the higher-ups would not permit them to get shabby. You never know when you are going to have to go fetch an admiral or ambassador. Besides, in the 23rd century, carpets last and last and last...
I'd just put down your basic thinned white glue and do the gray flocking thing without any attempt to make it look bad. Starfleet stuff has a sort of candybox too-perfect look about it anyway.
If it were a Klingon ship the carpet would probably have a lot of Pepto-Bismol colored stains...
WinBear