Help! Battery or electrical or keyless entry problem?
 

Hi I've been having screwy electrical problems: my factory alarm will go off every now and then for no reason which bothers the heck out of my co workers.

Yesterday after work, I used my key fob to unlock the door as usual, and when I turned my ignition the alarm went off, then the engine won't start anymore, making click noise when I try to start. All lights will still work.

After calling AAA, they tested the battery, seemed fine but drained below the normal voltage, so he charged a little bit and started the car. AAA guy suggested something is draining the battery little by little which can cause the alarm to go off.

Today I noticed when I removed my key from the ignition, the odometer will still shows the mileage. Doesn't the odometer goes off once the key is removed? This leads me to believe the electricity is still used somehow even the key is removed in my car, thus the low battery voltage.

Does anyone has any similar situations? or how to diagnose to see where the problem is that causes the battery still in use (drain slowly) even after key is removed?

The car:
2003 Camry SE 4 cylinder w/ factory keyless entry

Thanks in advance.