I have found Lipo's are the battery to use in situations like this. The issue is more the battery then the ESC. If you use a lipo this issue will stop or fry your ESC if the motors are truly drawing that much amperage.

IN all the cases where I had tanks shutting down from to much amp draw( Brownout) the Lipo cured it every time, and to date all my RX18's are still fine.

Also what gauge wire are you using to supply the ESC from the battery.