WE GOT IT! 🙂
A couple of things...
First, for some odd reason, when I typed in the number manually using the keypad, then "Call," it dialed and went through as expected (press and hold the * key to queue a "pause"). Previously I had been typing the number in one of the number fields in the "New Contact" area, which wouldn't even dial. After calling it once after queuing it manually with the keypad, I was then able to use the +person icon and save it that way. It will dial properly now.
Secondly, I had also been testing using the speaker so I could hear what was happening as it was sending the keystrokes. Turns out the mic was picking it up, so I was sending feedback tones as well which was confusing the voicemail system.