There are going to be additional SATA ports on the motherboard. Just get yourself an extra SATA cable, plug your old drive into one of the unused ports, plug it into a spare power connector and just login to the Windows 11 installation on your new computer and transfer the files over like it was a USB drive.
Did you enable it in NVCP for both windowed and full screen games?
Did you turn off V-Sync in Cyberpunk and Jedi Survivor?