I would opine that for there to be enough heat running around to be transferred from one barrel to the other and cause a cookoff (in a plastic shell), the barrels would be so hot the shooter would not be able to touch them without burning himself. My experience with cookoffs actually happening (as opposed to being talked about) is limited to the M60 MG, and getting to the point of risking cookoffs there came only when the barrel was close to glowing hot.
I suspect your problem is mechanical, not thermal.
