It's probably not asbestos.
But if it's a '96 you might want to replace the mattress and bedding if they are original. Also any carpet. Then look for mold/mildew. Check all the wet areas, shower, under sink, toilet area. If you discover the blackish mildew anywhere, then mix up a 50/50 mix of water and Clorox and spray the area thoroughly, caution do not get on any fabric.
Then, don't overlook any roof/wall areas where you may have had a leak. If you have water stains on the ceiling, then you may have mildew growing between the headliner and the roof. Check it all out and hopefully you will find the culprit.