Broken legs can't be good.
I suggest that you first download the manual for the machine.
http://arcarc.xmission.com/PDF%20Arcade%20Manuals%20and%20Schematics/?M=Dis just one site. Google for "dig dug manual" and see what you need. Download the schematics.
Next, you have to learn a little about the boards. Learn a little generic board repair and all about the logic probe. Mowerman has a good site on how to use a logic probe. (they are like $15 bucks at radio shack and other electronics places)
http://users.adelphia.net/~68hc11/ for the mowerman files, good reading!
Then look over this board repair guide (it's for pinball, but it is the same for any board)
http://www.marvin3m.com/begin/index.htm If you want parts, here's da man :
http://homearcade.org/BBBB/Bob Roberts. He has all the chips. Even if it isn't listed, tell him what you want.
What I would do is look at the board first, as you have done, then identify chips that look bad first. Take a look at the schematics and see if you can find that chip. Dig dug has a very extensive repair manual that includes trouble shooting. You are setting off the watchdog circuit for some reason.
You can go and find old files on the RGVAC (rec.games.video.games.collecting) group. There's years of advice in those logs. You can also look up arcade repair logs on the net. That will give you more clues to what to look for. You did the basics already.
Next you can just replace the chips that are what you would consider bad. Or you can get a logic probe and hook that board up to a power supply with 5V. (an old computer power supply has that). Then use the logic probe to check each leg. Compare that readout to another chip like it. If it's different, then you may have a bad chip.
Buy the chips you need from Bob Roberts or a hundred other guys who sell them on the net. Look for "arcade chips".
That's what I'd do.
Failing that, just get on the RGVAC and ask if somebody can fix it for you cheap!