Can you check in the manual and see what fuse F3 on powerboard is for? This will help us track it down..
OK. The schematic shows that fuse F3 takes 25V AC from the transformer and puts it through what looks like a bridge rectifier (please confirm, on the schematic it is drawn as an array of diodes in a square with a 22 Ohm resistor in the middle). it then is fed through to the playfield as 2 things: SOLENOIDE SUPPLY and SOLENOID SUPPLY [BOOSTED]. The testpoint for this path is T.P.3. Looking at this schematic, and funny how this circuit is playing up, I wonder if a previous owner knew, or thought they knew what they were doing; someone has written on the schematic and the resistor in the abovementioned array has a '-' at one end and a '+' written on the other end. The same pen then wrote 'R2+' at the SOLENOIDE SUPPLY [BOOSTED] end and 'R1-' at the SOLENOIDE SUPPLY end. I dont know if any of that is significant. 
Also, I have checked the switches and all are OK. The only glaring thing I noticed is that the coil which fires the right sling-shot will not allow the pin to move down it. On close inspection, taking the coil out, the coil sleeve has melted and imploded into the cavity, hence the pin CANNOT go anywhere.
Phew, hope this is helping you guys, and THANKS, the response has been overwhelming and I really appreciate it. 
Did you replace the transistors? TIP 102 is NPN and Tip 122 is NPN 8A, should be able to use either, even if the amp rating varies from 5 - 8 amps should be suitable for this application? Did you replace the slingshot coil? Are any other coils burnt or not free?
