In a non-RO save, MechJeb doesn't seem to actually shut down a stage before separating it, leaving it with a running engine. This happens both after coast and if the next stage ignites right after the early shutoff one. Even after coast, the engine will throttle up again right before stages are separated. This makes it hard to use solid upper stages, particularly ones large enough to not require a coast stage before (which tends to give enough time to manually shut down the engine).

I'd suggest killing all currently active engines on "early shutoff" stage burnout.