Are your wires contained in conduit, and is that conduit attached to the box? Are the boxes the low voltage type with the open back? If low voltage, with open back, I coil a few loops in the wall, behind the box and once the drywall is finished and painted, I reach through to my undamaged wire.
If it’s just a conduit drop attached to the box, and there is a way to coil some wire in the ceiling that is accessible, I use nylon pull string to the box, and I don’t worry about that being damaged. I pull the string that feeds the wire into the conduit from the ceiling after drywall is finished and painted.
Worst case is all conduit, back to the head end, attached to the box and no access to ceiling, meaning if the wire gets damaged you’ll be pulling a new wire. Pull a nylon cord through, do not pull the wire until after the drywall is installed.
I've used steel mud rings too, and coil wire in the wall higher, or lower, than the ring so anything penetrating the wall through the ring, like a Rotozip, saw or knife, won't be in range. I use a heavy wire to hook my coil if I can't reach through to the wire.