1st, try reseating the screen cables. May be the touch screen cable is loose. You can also try to clean the connectors using a cotton swab dipped in 90% isopropyl alcohol. Once done, reboot and test the iPod.
2nd, you need to try another (known good) screen, in case the first one is defective or the touch screen cable was damaged during the first screen replacement attempt.
3rd, if none of the above solved it, you need to examine the logic board using good magnification, looking for anything torn, broken, damaged, part-removed from the board. If you find something suspicious, post a good picture here.
4th, if you find nothing as per step 3, it could be that the drop damaged some of the chips' solder balls between the chip and the board. This type of damage is quite difficult to repair, and quite expensive. At this point, it could be better to pay and have it replaced at an Apple store. But you need to put back the old screen. Apple will not replace a device that's been tempered with, even if you're paying to replace it.