Since the bed is shorter than your sheets.. how about just tucking the extra length underneath the mattress? Or using a sewing machine to modify a set you like, or using safety pins or such. Then you can keep your favorite sheets.
Our bed is technically a full size plus we have a very thick foam mattress topper that gives it more height and comfort, but we use queen size sheets on it. I love having a larger than needed top sheet at the very least - helps from keeping your bed partner from stealing them as easily :)