I have resolved the issue (which sound identical to your issue). I have been trying to fix this issue for 2 months. I replaced literally ALL of my components including mainboard, motors and screen, and nothing fixed this issue.. I plugged my SD card into my spare ender 3-v2 and that printer starting having issues as well! Turns out, the EEPROM.dat file on my SD card was corrupted. Put the SD card into your PC and delete that file, then make sure you're printer is OFF, insert the SD card, then turn your printer on.
I had a power-out whilst printing, which corrupted the SD card. This caused issues specifically with the z-axis not moving while printing. Otherwise I could move the z-axis to home or jog it on the screen.