Very glad to hear you're getting up and around a little bit! That's encouraging news. Less encouraging though is what you're seeing...

The unraveling is not likely going to stop on its own. I don't know why bands are still sold that do not have the ends sealed. Due to oil soaking, it is not a sure thing to fix. but here's how I'd procede if I was in your shoes:
1. Kevlar does not soak up oil. It will wick it because it is a weave, but as a fiber, it will not soak up anything. This may work in your favor
2. I would release the adjust nuts such that the band ends come free from their attachment while in place
3. Press pedals HARD for awhile to let oil squeeze out and then let car sit for a week or so to let as much residual oil drain off as possible
4. pack rags around bands to soak up chemicals and spray the ends with a brake cleaner and vacuum off excess. Repeat a considerable number of times. When fully dry, but without undue delay, go to step 5.
5. Separate bands from drums with some physical barrier that will fit. I don't know if wax paper would work, but you could experiment. Apply a gel form of cyanoacrylic glue along the frayed edge
6. Give it a day and trim off straggler fibers
This is not a proven fix that I have tried, as I've never been in this boat, but I do know this is (generally) how the bands are sealed by at least one supplier (sans oil, of course).
I know that cut ends have also been successfully treated with Glyptal or a substitute, but I don't think it would dry/cure like you'd want in the presence of trace oil, and I suspect the cyanoacrylic will...
best of luck regardless of how you chose to tackle this...
