If you can find Full or Double bed sized sheets (54 X 75) that do not have defined corners sewn in and full elastic around the bottom edge, you will have plenty of sheet under and around the larger but thinner mattress. Most sheets nowadays are made for mattresses that are as deep as 18 inches, which adds a lot of material. I cannot tell you what linen makers sell these type of sheets, but most of what I have found constructed without corners and full bottom elastic are microfiber, not cotton.
@Sarah977 has a very good suggestion for altering flat sheets too. If you are in a pinch, I would just make up the bed with two flat sheets.