well im not sure if it is a common thing but i got a 96 xlt and its just the arm rest but it was kinda comming off on the bottom and i was cleaning and i was scrubbing under there and it started to peel off i thought no biggie it will go back on well after much frustration i came to the conclusion it is probally not suppossed to do that i think it is suppossed to be attached to the black part under the vinyl and foam i pulled off
yes, it is common for them to start coming apart at the bottom. i have recovered several to fix this. from what you are describing, it sounds like you will need to take the door panel off, unscrew the plastic backing plate, then reattach the foam/vinyl to the plastic. this could be done with a contact cement.
i don't believe in using dye on large pieces. everything in those pictures is vinyl, including the charcoal grey door panels. you can see the original light grey plastic on some of the area, the rest is wrapped.