[TAKPAK]

The battery voltage would drop to/below 12V. Once they recharged, the slide/leveling system would operate for 10 seconds and stop again.

Part of the key here too, is what you are saying about the battery voltage. As the voltage goes down, the amp draw goes up. Could be a battery problem, or could be "line loss". Fully charged batteries without any load should be at least 12.6 volts or a bit higher, if they just came off a charge cycle. They should have enough "oomph" not to drop very much when in use, at least at first.