The ADS1115 did the trick. Not only did it get around my "one pin short" issue, the analog reads are now rock solid too.
I redid the "shield" incorporating the ADS1115 and everything works really well now. I consider the hardware to be officially complete at this point. I'm still tweaking the software a bit but that's pretty close too.
I have also started to write a Python script that will make the creation of trivia question cards, well, trivial. I see the light at the end of the tunnel.