no they should cut out as soon as the game is done. You have a shorted on transistor for the flipper relay.

Find what Tip 102 controls the relay. Ground one one end of an alligator clip to the frame or screw of the board, and touch the metal tab of the correct 102. the relay should click and move when you ground it out. look in the manual to see which one it is.

K1 does sound like the suspect. It didn't click at all when you grounded out it's Tip 102? Should hear it click over every time you touch it's tab.

it shows what TIP 102 it is in that part of the drawing you showed a picture of. it's Q something, but I cant' read it because it's all smudgy. Right after the 7402 chip, that transistor after that, that's the one that controls K1.