I think robertmee is on the right track here. You have eliminated the CPU board by removing the ribbon cable between the power and cpu board. I would be testing that board transistor next. I'd also be wondering if there's something in the wiring that caused both transistors to get shorted.

You're definitely doing the right thing by powering off that machine as soon as you discover the coil issue. Also, if you replace one transistor, may as well fix both boards at same time

In ohm mode test impedance from the center lug to the ground pads or screws on the board. You get what? And compare it to a working one next to it. Then remove the solenoids connectors from the power board and test again.

Yes. Usually when you get a shorted transistor you get 0 ohms. But 1.1 is way too low. But the part after pulling the plug doesn't make sense. I'm not sure what the plug is. The solenoid connector into the board is what I'm suggesting you remove. Try to isolate the board from the wiring.

Or better yet, disconnect the power board entirely and out it on your work bench. Then retest the transistor tip to screw plate. Is it still 1.1? If so, replace it. After replacing it, test it again same way. Is it like the others? Reinstall everything. If the solenoid still fires up locked on, test the transistor again. If its now at 1.1 or shorted then you know something in your wiring is blowing the transistor. I guess we should start there and move forward.

And with the board out, the transistor shows 1.1 ohms from top to ground? If so replace it.

Sorry but I'm not sure how those test and I don't have a board here to look