Lucy,
This is the way I do it, (window cleaning)
I wear a belt with two pockets; my right pocket holds all the wet stuff, t-bar, squeegee, old towel type rag. My left is the dry side, scrim, scraper, etc.
I keep any old towels and cut them up and have a box full in the boot of my car, scrim seems expensive to me, so I intend on make each piece I buy last.
I use the old towelling to wipe my squeegee dry and for detailing, I only use my scrim to polish the glass and remove any squeegee marks.
My scrim stays dry all day long.
Peter.