Is the sheet for a full/double or was it a queen sheet? Queen will be way too big. Assuming you have the correct full sheets, if you have a typical mattress it is probably only 12” thick, measure how thick it is. I shouldn’t say typical, because I need a 12” for my platform bed and it is usually hard for me to find, because most mattresses seem to be 16” or 18” now.
Do you mean you only want the elastic on the corners? I would think there are still brands that have that. Usually, the better brands are elastic all the way around.
You could get the better bedder and the exact fit of the sheet won’t matter. Or, get a flat sheet and make your bed the old fashioned way with hospital corners.
Edit: I just googled and there are still sheets that are 12” pockets. I haven’t bought sheets in a long time so I can’t speak to which ones have elastic exactly how you want, but if you go to a department store that has bed displays you can look at the sheets or a salesperson can help you.