I hear from people with this condition pretty frequently and I'm surprised it doesn't get discussed more on the bulletin board (dryeyetalk.com). I don't have any info on changing sleeping habits - that's a toughie.
But the products I'd tend to recommend first for this would be:
1) post LASIK style goggles (like the simple sleep goggle I have in the shop - only the picture is now inaccurate because they don't make that one anymore) - the point with those is it's got a rigid front with a gap in front of the eye so you CAN'T rub your cornea on it no matter what you do during the night, and it's hard to remove. - They are not the funnest things to wear, for sure, but they work.
2) Expensive but maybe better for your case:
Medtronic NITEYE bubble eye bandages. Never used them myself so I can't comment on the type of adhesive they have but they're basically a plastic bubble over each eye with an adhesive. They are single use and expensive. But every once in awhile I run across someone for whom it's been just exactly the right thing. I don't sell them because I can't get the manufacturer to sell to me but here's a link:
Depending how gentle the adhesive is, that's probably what I'd be trying if I were in your shoes.
One other thought - bear in mind I have NO knowledge about this for your specific purpose (never heard of it being done, and for all I know there might be medical reasons against it) but it's something that came to mind and it might be worth asking your doctor about
You know how people with poor lid closure get gold or platinum weights implanted in their eyelids? Well, I read awhile back about a product that lets you put a weight on with an adhesive for use just at night. Here's a link:
If your lids evert very easily I just wonder if this might possibly help them stay put. What I'm thinking specifically is that if you had one of these on the lid while ALSO using a really good protective goggle or bubble eye bandage at night, maybe somehow it would help 'retrain' the lids.
edit... On second thought the weight might backfire (the weight might pull the lid back depending how your head is oriented). Need dr. input on this.