It is entirely possible to alter what you have to create a full size fitted bed sheet.
Buy a full sheet set that matches the twin set you have.
The length of a new mattress will be the same as your twin mattress. So, if you cut off both the left and right sides (plus seam allowance of an inch) of the fitted sheet and then cut the twin flat sheet to sew it in the middle of the two cut sides, you maintain all four corners of the fitted sheet for the full size mattress, while keeping the seams on the left and right edges of the mattress. The full size mattress is 54" wide and the twin is 38" or 39" wide, so the twin top sheet will certainly be enough material to create a bigger full size fitted sheet. Now, with the left over sheet fabric you will have from cutting the top sheet, sew it to the top edge of a new solid top sheet so you will have a matching set.
This way, you can mix and match the newly altered sheet set with the new sheet set. The only difference will be that you will only have one top sheet between the two sets.
If you plan ahead and cut the top twin sheet carefully, without waste, you may even have enough left overs to trim out the new set of pillow cases to create a mixed and matched set.