Just cause a LiPo isn't swollen doesn't mean its good. Even put in storage mode they start to degrade. They lose their ability to hold a charge as you are seeing now. If your charger shows IR then see what they are at. A good battery should be between 1-5ohms. Anything above 20 should be retired. At 30 or higher you run the risk of fire. If it is a standard 2s 5000(ish)mah with about 5 min of runtime I would imagine the IR is quite high. Run at your own risk. Personally a 20-30 cheap battery is worth it if I dont end up burning down a house or the car its in.

Also a lipo is low (should not be run lower then) at 3.2 a cell so for a 2s battery its 6.4. Your ESC could have the cutoff set at 3.5V per cell or 7V. Another contributing factor is that since they are so degraded that the voltage drop during use hits the LVC but when there is no load it jumps back up a bit. This effect would be worse with a older battery.