Door Sensor with a Few Problems

I have a first gen alarm system and several door/window sensors. One gives me a low battery alert every few months, while other stay good for a very long time (and I haven't even needed to change their batteries). This is not a primary door, and others, including our main front door less than 10 feet away, don't have the same issue. This last time, I used a battery tester on the battery, and it shows the battery being full, but this sensor is still showing it as low. Additionally, sometimes closing the door makes this same sensor register as being tampered. It's very difficult to get it to reset and see itself as resolved. I've removed it as failed a few times (it's also frozen and failed to reconnect in the past), but that hasn't resolved the issue. Any other things to try, or does it sound like this sensor is failing? I do have the Professional monitoring with the extended warranty. I just don't have time to call in and talk to support right now, and they don't give any other support options...