Under Settings, select Devices > Bluetooth Devices and hover over your controller name with drift issues. On a working PS4 controller, press the Options button in the upper-right corner of the screen.

Fortunately, the flashing blue lights on the PS5 controller are an indicator that it's in the pairing mode. This is normal behavior if intended -- say, when you're looking to pair your own ...

The notorious Xbox Elite controller bumpers can and do fail, but there's a real easy fix.