Ya your gonna need to provide some more info if you want to get some real help.

What are you saying, the Game Over Relay energizes in the middle of a game?
If so, check for vibrating Slam Switches. I think there's one in the rear of the machine,
and maybe in the coin door..

Long back, I remember working on machines with the double Pin Motor Reset cams..
What it sounds like is happening is the Pin Reset Relay is energizing, and then denergizing
without a (suppose to be) closed switch off of one of the Pin Reset Motor's cams taking over
the responsibility of the Score Motor's circuitry..
Once the Pin Reset Relay energizes and the Pin Reset Motor begins its revolution, the Pin Reset Motor's
Run Out switch should make (touch), which also completes the Score Motor's circuitry, then
another set of contacts off of one of the two cams opens cutting the Pin Reset Relay's circuit..
With the machine on, you can manually activate the Pin Reset Relay which will energize the
the Pin Reset Motor and when the Pin Reset Motor stops, you can study what set of contacts should
be making (touching) to continue the Pin Reset Motor's circuit. Once the Pin Reset Motor completes its
revolution and the pins are reset, those same two contacts should open and put the Pin Reset
Motor at rest..
Later, we can address other problems that the machine may have..

I know that many like jumping right to it and try playing a game, but going through the switches and
units (or at least most of them) before plugging it in is really the way to go. Then it'll help to answer
questions way before they're even needed to be asked..
I haven't worked on 60s William machines for a long time, but I'm pretty sure that the pin unit
switches under the pin hood energizes the Relays in the Relay Bank. It appears that the #1 pin unit
switches are energizing the #1 Relay in the Bank, but the other pin unit switches are not. It also sounds
like the #1st Shot Relay in the Relay Bank is not energizing. The last row of rollover switches (on the playboard)
is what energizes the First Shot (also 2nd Shot) Relays..
It's a little surprising that 9 of the 10 pin unit switches aren't making good contact, but you'll first want to tighten
the switch stack screws that are under the pin hood that the pin units function, then clean and properly gap then.
If you have the schematics, perhaps you can take a pic and post..

I have to say they are a little blurry, but can be made out for the most part..
Yes, the Pin Unit's switches do energizes the Pin Relays in the Relay Bank and
not vis-versa as of the earlier United machines..
It seems a little unusual that many of the rollover switches are that much out
of adjustment or need cleaning, but possible, so snugging up those switch stack
screws and then cleaning their contacts wouldn't be a bad idea..
You cleaned the plugins, so I'd try that.
I'll recheck the schematics, but I'm pretty sure all the other pin units primary
circuitry is the same as the #1 pin's. The only difference is the pins individual
rollover switches..
Also, each pin unit (under the pin hood) has a make-break switch. When the pins
are down, The two contacts that are making (touching) completes the circuit to
their own pin unit coil, then once the pin is energized (up) those contacts break
so that its coil doesn't remain energized, and the two contacts that are now making
are to complete the circuit to the pin relays in the Relay Bank. Once the pin relays
1 - 10 (in the Relay Bank) energize, each one has a set of contacts that break
to discontinue it's circuitry. Before the pin relays in the relay bank energize, those
contacts in each pin relay have to be making good contact so that the pin unit's contacts
can complete the circuit to the pin relays (in the Relay Bank) circuitry..
On these bowlers, there can be quite a little bit that needs to be checked out..
And also, before adjusting, tightening up all switch stack screws is always a good idea..

Oh sorry, I got thinking. You had said that the pins are going up, and then
all goes dead. So the rollovers are doing their job, but still the rest of the switches
need to be checked out..
Yet maybe the last set of rollovers could use some checking, but the 1st
Shot Relay not energizing is most likely in its circuitry, but we'll first concentrate
on getting the Pin Relays to energize..
Also, nothing happens after the first shot, but does the 1st Shot Relay (in the Relay Bank)
energize?