I just ran into this problem on my Shadow pinball machine. Some things to look out for:
1) Are your batteries installed correctly? If they're in backwards, the game won't save settings. Make sure that the positive terminals are all facing up.
2) Are you 100% certain your battery holder is in decent shape? William's used honestly garbage battery holders. They might look alright, but aren't making connection. You can see if power is getting to the RAM to see if it's good.
Stick you multimeter in DC voltage test mode, stick the black lead on ground (I just stuck it on one of the screws that holds the board in the backbox, and touch the red lead to the last pin on the RAM (easily identified, since it usually has the big "Warranty Void if Removed" sticker on it). The last pin (pin 28, I think) should be the very last pin to the right of the "notch" on the chip. See if you're getting voltage there. You should see around 4 or more volts.
Also check for power at Diode 2 (D2) (its a small glassy looking diode to the right of the board, it'll be labelled). With the black lead still grounded, one side of the diode should be 4 or more volts, and the other side should be around 3.6 or so volts. Anything else, and the diode has failed.
After all is said and done, I=ideally, you should install a satellite mounted battery pack (or NVRAM) to keep the MPU safe from future corrosion, but if you're super lazy like me, you can sub the standard alkaline batteries for some fancy lithium AA batteries. They won't leak.
Good luck. I'm betting the issue is simple, but you shouldn't have an issue testing everything else "downstream."