Yes...after sanding and cleaning the existing particle board, you can apply Bondo (automotive body filler) to the crumbling edges. Once dry, you can sand a paint them to look new. You will not need very much so get a 1-quart size or less.
Bondo does make an "All Purpose Putty" that is similar to the "automotive filler". It comes in a 1-quart or 2lb. 14oz.
Net weight size. This will work fine for you.
You need to seal them good and repaint them. To seal the crumbling edges us some glue ,epoxy glue would work good . Lots of it that can soak into it to hold it all together .
Then paint the doors.
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