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.

Test each ballast outside of the fixture, using separate power cable / incandescent lamp / fluorescent lamp and starter. There is drawing in my post on the previous page

I'll keep this page bookmarked.  I'm mad at these signs today, and don't want to look at them for any more time. And there is no skin left on my fingers from so much wire twisting.  lol

 I sent you a diagram for dual preheat circuit, dont know how to post drawing on lighting forum.
    this is with the emegency ballast remove from the equation. once this preheat circuit set up and working
  then pop that eballast from the second exit light emergency ballast that sound like is working. into the circuit with the starter block and FS5 from remove from preheat circuit. then will go from there
with the eballast problems