I know someone will flame me for this suggestion but!
Those solenoid used replaceable coils ....well back in the late 80'S they were pretty common. Anyway they are relatively easy to take apart. Once apart if you ever took a close look you will find that the coils are fine copper wire and if you check the resistance ..well just start unwinding the wire until you find where it has broken. If you are lucky it will be in the first couple of feet,maybe five!
Try applying 12 volts to see if it works without the removed wire!
This doesn't always work ,but it did for me about ten years ago and that old water heater is still working today!
