A garage door is under spring tension that can move fast and hit hard, and a cable under load is no different. We don't recommend homeowners replace cables themselves, and we won't walk someone through it over the phone - not because the job is a secret, but because the door has to be secured and the tension released in the right order before the cable ever comes off the drum.
If your door dropped crooked, the safest thing to do is leave it alone. Don't try to force it level, don't cut the slack cable, and don't run the opener trying to get the car out. Running a motor against a jammed or unevenly loaded door is one of the more common ways a small cable problem turns into a bent track or a cracked panel, and on a door this age, a cracked panel usually means the whole section has to come off and a replacement has to be sourced - panels from 1960 aren't a stock item anymore.