Well guys, bad news, bad news. I got the second sign today (which I will refer to from now on as "Alkco #2"), and I first took the newest-looking white ballast from Alkco #2, and put it into Alkco #1. I got a brief flash of light from one of the bulbs, and that was it. They continued to make their same old pinging noises. So, I spent about an hour getting Alkco #2 re-wired correctly by following the way Alkco #1 is wired. I hooked the white ballast back into Alkco #2, powered it up, got a brief flash from one bulb, then it died. The same as what just happened with Alkco #1. Both bulbs refused to light, however Alkco #2 does not make pinging sounds like Alkco #1. Now when I rapidly push the test button, both bulbs will briefly, dimly light from the power surge. Other than that, they won't stay on. Now, just to eliminate all doubt, I'll take the original brass-colored ballast from Alkco #1, and put it in place of Alkco #2's white one. Now if it doesn't work after that....then I'm screwed. lol Because I know nothing about circuitry.
I'd have to just either deal with the fact that they don't work, or just place a generic fluorescent strip light into one of them.
Now I've got a whole box of T5 bulbs at least, for future use.
EDIT: Swapping ballasts left and right did not work. Bummer. I'm done with these signs for awhile. There is like...no skin left on my fingers from twisting so many wires together. lol I'll probably mess around with them more later, in a few months maybe.