I also have a Williams sound board with a very loud hum all the time in attract and game mode. Currently looking into this. I tried swapping speakers as well with no change. Speaker frame is grounded and all mounting screws are present in the sound board. Voltages are good as well. The "volume" of the sounds changes with turning the volume pot on the board, but the loud hum never goes away. My board is in a Flash.

Problem solved. I remembered I had a similar issue on a Bally Playboy. There was a constant hum that would get louder at times. On Playboy, when in attract mode, a feature lamp lights and it displays the high score to date on the displays. When this would happen a very loud hum/buzz would come out of the speaker even louder.

The solution was to isolate the sound board from the mounting brackets with small nylon washers.

I put one between the board and bracket and one between the screw and board, to sandwich the board and isolate it.

This completely eliminated all loud humming at all times on the Playboy.

I just did the same thing with the sound board on my Williams Flash and got the same result. The humming was completely eliminated. Perfect!

Obviously, it's still a good idea to recap the sound board if they are original caps as well, but isolating the board was the solution to eliminating the loud humming in this case.