You'd definitely have to keep them covered with mulch, if you want them to break down, as this will help it stay damp.
Cotton is hydrophilic, meaning that it draws
water, which can be a good thing if it is kept wet, but if it is not kept wet it can get wet, draw that water in, and then release that water slowly as it dries and do this many times, as it seems to like working with water. However, if it was wet, and mulched and the soil dampened initially beneath it, then the mulch
should keep the cotton damp, and that should eventually cause bacteria and fungi to get into it.
I've seen plenty of fungi/mildew/mold-ish stuff happen with cotton when I've found it in old abandoned homesteads/collapsed buildings; so it will take on living communities when it is consistently damp.
When I was a kid I found a rag on the driveway, that was damp after the rain, and I picked it up and it was teeming with worms underneath... and had a worm farm, so what I used to do was take old towels, soak them, and then lay them on the grass in the evening, and at night the worms would come and hang out underneath the moist towel, and I would pick them up in the morning and put them in my farm. Worked a charm. So, I wouldn't worry about the cotton being a problem with your soil community... so long as it's damp.
You could cut holes into your sheets, at regular intervals (say a series of 6 inch holes with a one foot solid area of sheet between) and mulch heavily around the holes, and plant in those holes exclusively until the sheets breakdown; this would allow you to concentrate weeding on just those holes and the sheets would do the job, long term of suppressing weeds. When they finally break down, you can plant in the alternate locations.