Dave,
I do pretty similar.
I cut a strip as wide as I want the patches, dip them in deer tallow and let them cool. Then I cut them near through with scissors, just leaving them attached at one corner, so as to make square patches.
I have a short piece of leather on my hunting bag strap, with a metal hook on the end, and the length of patching is hooked on and hangs down. V easy to tear the last one off the bottom. A dozen patches doesn't get in the way at all, and I keep more rolled up in the pouch pocket.
Seems no need to wipe between shots. If V hot and dry, a bit of spit now and again on a patch keeps things running no bothers.
Richard.