Door locks not working properly

About a month ago, I noticed that my car started to chirp and unlock 5 times in a row whenever I used the proximity sensor on the handles (both driver and passenger side, but not the trunk). Shortly after that, the power door locks stopped working unless the car either was on or had recently been on (within 10 minutes of turning off, maybe--I never timed it).

I just replaced the 12v battery, which was entirely drained despite having been purchased in April of last year, and the same symptoms are still present. In addition, my car no longer beeps when I lock it. The key fob batteries have been replaced recently and, again, the unlock/lock function works as normal both with fob and with the power locks whenever the car is on or has been on recently. The manual lock also works fine.

Is this a problem with the actuators? A relay? Something with the proximity sensor specifically? I'd appreciate any insight--I would think that I'd have to replace the actuator motors, but it's odd that both sides would stop working at exactly the same time and retain normal lock usage whenever the power is on.