I would unplug every connector I can find and write them down in a log book. Let them air out for a few days and re-connect everything. I would also swap out the battery regardless. A lot of times those battery testers do not diagnose a bad cell.
Pull the carpet up and check the ECU for wetness. Also scan the ECU with an OBDII reader and see if any CEL codes are pending.
With the car off, use an ammeter and pull your fuses. Go to each fuse holder and complete the circuit with the meter leads. Check for excess current. If you find something post the fuse number here and at least that may narrow down the short to a small group of sensors.