For what it's worth, I don't think it's an endemic problem with older Winnebagos. My 2002 Suncruiser has what I think is the same material (a textured, fuzzy fabric) and it's just fine except in the bathroom where it had a skylight leak. I'm in Sacramento but my MH started life in FL, then CO before moving to Sacramento where I purchased it, so it's experienced the full spectrum of climates without an issue.
I replaced my bathroom ceiling with FRP (fiberglass) paneling from Lowes. I don't know if, short of replacement that there's anything you can do, although it might be possible to stabilize it with some sort of paint. This might be problematical since the glue holding it to the Luan is water soluble. When I removed my bathroom ceiling fabric, a spray bottle of water made it very easy to remove.
Replacing it with FRP might be your best bet. It's not difficult, especially in the larger areas. It comes in both 4 x 8 and 4 x 10. The 4 x 8 sheets just might be wide enough (worst case, a little molding along the sides would work) and they're much cheaper at $44 vs $76:
https://www.lowes.com/search?searchT...log=4294518226
I know you wrote you weren't up to DIY but the FRP installation shouldn't be too labor intensive/expensive. Alternatively, you could install some form of painted plywood or other material.