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My experience is that the dome light bulbs burn very hot. I've burned my fingers on them.... very bad ouch! I suggest you make sure the contacts for the bulb are properly set in the dome light base. If the contacts are too low, the bulb will be very close to the lens which could cause it to melt. You can remove the base, pull the assembly down, look at the top and make sure the contacts are positioned correctly in the base. They clip in the base a certain way, but are easily re-positioned if the are wrong.
While the old original covers are indeed brittle, broken tabs and yellow, I've never had one damaged by bulb heat. Only the replacements seem to melt. Cheaper plastic? Nah....that can't be it.