Rewire outlet without drilling through drywall
The best way I have found is to break the old box out and remove the staple if possible and possibly use the old wire for a pull but not often will this work. If it won't I  will drill a hole from the attic or crawel space and fish the new romex in. I use an old work box because you can't really secure the box with nails , the old work box hangs onto the sheetrock or plaster. No staples are required behind the finished spaces but they are required in the attic if not run through holes. Have done many old homes this way that were permitted and inspected .
Glow sticks are helpfull (1/8" or larger fiberglass that glows after being in the light) and a coat hanger formed to grab the end of the stick and pull it to the hole and then the wire connected to the end