Alternator or wiring is toast.

You can drive on the alternator alone if need be, as long as you do not get close to the rated amperage on the alternator (wouldn't run more than 50A on a 90A alternator for example).

If you are losing power when everything is turned off, then the alternator cannot produce enough power to replenish.

Would charge the battery up overnight, start the car and have the headlamps on, then check the alternator output @ the alternator terminal itself, then @ the battery and see what the difference is. IF it is around 14V @ the alternator, but around 12-13V @ the battery
then the battery lead is breaking down. If the output is low@ the alternator, then that is the problem.

You need to do it with the headlamps on really as this removes the surface charge from the battery to help prevent a false reading, plus makes sure that the diode packs on the alternator have kicked in sufficiently for an accurate test.

An even better way would be to use an ammeter inline to check more accurately, but as most people do not have ammeters rated to 90A+, the method above will be suitable enough.