I'm not sure about normal actions on Ship Mates but assume that it works like earlier Gottliebs where the hold relay doesn't normally engage until a coin is dropped or the replay button is pressed. If so, usually the startup routine involves a switch that closes (and is associated with the coin relay or replay button relay) that then completes the circuit to the transformer which then causes the hold relay to engage. And so, perhaps one of these switches is closed when it should be open.

Good luck!

I agree with @tfduda. This is the H/30 Volt Hold relay circuit:
Ship Mates H relay (resized).jpg
The schematic is typically drawn with the game reset and then unplugged. When the game is plugged back in there should be no way to fire the H relay until the SB/Start relay fires. It sounds like the H relay switch on the right side of the schematic is stuck closed or shorted.

The reason your game resumes mid game is because power is getting to the playing/scoring circuits that would normally be cut off by the H relay switch until the H relay fires.

/Mark

Oops. You probably meant the R/115 Volt Hold relay, not the H/30 Volt Hold relay.
Ship Mates R relay (resized).jpg
That shouldn't fire unit 120 Volts gets into the transformer which shouldn't happen until the K or W relays are fired by a coin chute.

The first think I'd check would be the three parallel switches that can get power to the transformer on the K, R and W relays. All three should be open. One of them could be stuck closed, shorted, or have bent tabs, a solder blob, etc.