We had the exact same thing happen. I finally took a small screwdriver, heated it up with a small torch and melted the threads a little. It was just enough to keep them from falling off but not so they were too tight. Still had to snug them up once in a while, but at least they would not come apart and fall under the stove. Amazing how they could fall under the burners!
Later, snugging them up once in a while was solved by one half of the cover shattering on a recent trip.
When we opened up the camper, the stove was full of bits of safety glass. It was a mundane trip (<50 miles), no rough roads that we recall, no reason we could think of why it would shatter. It was pretty easy to just sweep them under the burners for the rest of the trip. Used the shop vac when we got home to clean it up.
After that I used the remaining half as a template to help me fashion a wooden version with a jigsaw. A little sanding and some tung oil and it seems to work for now. I still have the center hinges and may cut it in half and attach it like the glass one.