So if I've understood your problem correctly, the game starts up and plays, but the tilt light is always lit on the backglass?

If this is the case, you probably just have a contact stuck closed on the tilt relay. The relays should be labeled in the cabinet, so you shouldn't have too much trouble finding it. Look for the tilt relay, and check the switches in the stack for proper motion and adjustment.

As for tilt mechanisms, games usually have 2 or 3 types: The plumb bob, the roll tilt (needs a ball to work, but 99% of the games I've ever had came missing it. Don't worry about replacing it.), and the slam tilt. These are switches with big weights on them. They're found usually on the coin door and on the playfield. When triggered, these will instantly shut the game off (unlike the "standard" tilt which just ends your ball in play).

There can also be one on the bottom board, same kind with big weight on it. if a tilt switch is stuck on, i would assume the game would not play, and the tilt relay would be locked on.
you can test the tilt by starting a game, and swinging the plumb bob so it hits the ring, instant tilt. depending on how that game works, either the playfield will go dead, until the ball drains, and the next ball will be served, or the game ends right there.
nice find!

If the game functions and the tilt light is just lit, that means that one of the switches on the tilt 'trip' relay that fires when you actually tilt is a little too close.

It should be Normally Open (NO) and it gapped at the moment so that it is Normally Closed (NC). You can confirm the switch by reviewing your schematic and matching wire colors (if possible).

If the game is in fact tilting out as soon as you coin up, then it is probably one of the standard tilts like the plumb bob or something touching something that it shouldn't.

Good luck!