Target, BB & B have a lot of XL sheets this time of year. I would just get the correct size and not make taking care of their new space that much harder.
I think you want to buy what they call "Jersey Sheets" Go to Bed, Bath & Beyond.
I would get XL sheets if they are XL beds. Then I would make sure to get a feather bed, egg crate, foam topper - something - to help with the horribly uncomfortable beds they have at schools! They're not too bad for a few nights, but to sleep on them all the time, you really need something on top if you don't like to sleep on a rock. And then the twin sheets really wouldn't fit!
I think my twin sheets may have fit since the mattress was so thin, but honestly, I had the XL sheets, so I used those. I did know some people who had regular twin mattresses in their rooms, and thought the XL sheets were just too big for those.