Is this the correct flasher?

When in test the A/C relay is also being pulsed on and off, that's why it flashes in test.

Find the resistor board with the black/red wire. This wire goes from the resistor board to the diode board. It might be pinched somewhere to ground under the play field. If you unsolder the black/red wire from the resistor board it should turn off.

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Normally I would have assumed that Q-25 was shorted but you say the 2a solenoid is working correctly.
If you replace the black/red wire and leave 1P11 off, does the flash lamps stay off?

I going to bet it is melted a bit from a transistor that is leaking current with the power on, but tests OK with the power off. I would replace Q-25 and its predriver. This should take care of your first issue.

Now for your second issue, the 50 volt power fuse.

First is why is it doing a ball search? No balls?

Put the 4 balls in the game. You don't need a fuse to troubleshoot. Turn on the game. Use a voltmeter and check the metal tabs of the TIPs for the snubber relays. A good tip will have @ 30 volts. If I remember correctly there is 6 snubber relays.

Here you go.

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So it looks like both issues are tied together. Time for a logic probe on U -28.

Both test points are on the cpu board.

Has the power supply been recapped? Check the 24 volt power with a voltmeter for both DC and AC.

You are getting a pulse on the 24 volt line causing the relay to pulse, since only the one relay is pulsing there may also be a issue with that snubber relay.

Playball!!