cj:

There is a device called the Over Voltage Protection (OVP) relay that is responsible for a number of strange symptoms when it fails, although keeping the engine from shutting off is a new one. It often has a red top, with a clear lid over it, and one or two 10A fuses under the lid. Check the fuses, and replace if melted.

The lights remaining on is a clue that power is being fed back, either through a defective relay, or through defective diodes.