Do you have any pictures of the mpu, particularly around the battery holder? Ballys of this era are notorious for battery damage. I doubt that the solenoids are defective. It's far more likely that you have issues on your PCBs. Luckily, newer versions of all of the dash 35 boards are available. Check out this site. I've had good luck with them in the past.
If it was me, I would just replace the mpu and the driver board rather than fiddling around with fixing the ones you have. Check your power supply out. Replacing it is more involved.
You're also fortunate that there is a lot of info out there in repairing these old ballys. Pinwiki has some great walkthroughs on fixing up these old games.
Power play is an awesome game. Good luck and have fun!