Yes it could kill your battery, especially if your new battery is not being fully charged because of a failing alternator. I would park your car on a level surface, start the car up so that it rises to drive height , then while the car is running exit the car and turn off the air ride switch in the trunk. Drive this way for a few days to see if you loose any air in the air ride (the leaky bag will show). It is not unusual for alternators to get wiped out by dead batteries.
I'd disconnect the battery and connect an ammeter in series from the battery to the battery cable, and check the current draw. Set your meter to the highest amps possible, so you don't blow its fuse.
Turn the air ride switch off first, and don't open the door during the test. My 94's draw about 100 milliamps for the first hour or so, then drop to about 38 milliamps.
If your car draws much more than that, I'd start removing fuses one by one, and see which circuit is drawing the current.
Did you ever get this fixed?? Take off the negative battery cable with the car off, then use a multi meter and put one lead to the negative on the battery then the other lead on the negative cable and it will show you how many volts are being used while the car sits