I took up straight razor shaving a few years ago, as well as the old "double edge" razors. Once you develop the skills, it's easy peasy, and I have less skin irritation using either versus a modern 5-gazillion-blade Gillete or Schick. I admit that I do get a closer shave with the double-edge vs. the straight, but I still pull out one of my straight razors every now and again just to be sure I haven't lost my skills.
But... back when this was a bit of a hobby for me I did a bit of research, and I came to the conclusion that you really should NOT treat strop leather. The "fuzzy" side is for using a polishing compound if you want to, and the flat side is for the final strop. If you have one that'd a bit damaged or has nicks in it, enough that you are not getting shaving-sharp edge anymore, and assuming that you are convinced that your razor does not need to be honed, my advice would be to retire that strop from shaving strait edge razors, and either use it for other knives where the edge doesn't need to be THAT sharp, or use the leather for other things (flint wrap, tooling leather, etc).