OK... So far so good after switching the DME relay with the Headlight washer relay (Blue relays in the Engine Electronics Box). Starting every time and no "auto shut offs."

I have also done some more reading.

The connection between the DME and Starter is to hold the starter in Auto Start. It's not a signal to the DME.

Also, the Engine Electronics Box has a cooling fan. It draws cabin air into the box for cooling. Like most electronics I suspect if the cooling fan is not working properly, neither will the electronics.

The fan has a niffty feature which makes it easy to test. On start up the fan comes on for a moment to prevent the bearings from seizing over long term. Pull the cover, have someone else start her up and the fan should come on then shut off.

If that doesn't work, the fan is supposed to have 12v to it at all times. The DME has a temp switch which gives the fan a ground to turn in on. Check for 12v on the plug to the fan. You can also put 12v to the fan to see if it spins.

The only way I can think of to test the temp switch is to heat up the DME in some way. I just can't think of a safe way to do that. Any ideas?

If the fan tests good and the problems come back, I can say it is an absolute, defininte, maybe that it's the Ignition switch.