demandingiowa:
Sorry, should have been more specific. The problem was solved by connecting the external power supply's positive lead to the VIN pin on the Arduino UNO.
How was the Uno powered before you did that, since it was 'somewhat' working? The reason I ask is that multiple voltage sources to a single item are usually not a good idea. With that, the 'stronger' (higher V) source will be pushing power into the lower V source.
When two separate items fail to communicate properly the usual fix is to common the grounds at one point.
So, my question is: why do I get a HIGH and a LOW reading when the outputs are all on?
-Either- because it's -cleanly- cycling between high and low very quickly, -or- it's at an 'in between' voltage that the probe can't decide which to display. With enough LEDs you might be experiencing voltage drop when 'a bunch' of them turn on all at once.