Yes, I have. This display, and all of the others have worked flawlessly without pullup resistors. After reading this, I figured I'd give that a try, but as I was wiring it up, I realized I hadn't connected the ground of the Uno to the display. I had only connected the (-) from the external power supply to the display. Connecting the ground from the Uno immediately fixed the problem.
When I came back to reply, I saw the response by CrossRoads asking about the grounds....yup, that was it!
Thank you both for all of your help. I will be buying the bits and pieces for a voltage regulator tomorrow so I can incorporate it into my current design. I will split the incoming voltage from the 9V wall wart between the soon-to-be-built regulator (which will send 5V to the displays) and the mega, so I don't have to use two wall warts. If that fails, I know I can fall back to the dual wall wart option.