Thanks for the response, guys.
enhzflep- That method is CLOSE to what I was looking for, and I did get some mileage out of it... Creating new vertices - good. Creating new vertices and then shifting or moving them to new locations- bad, as it also shifted the associated textures with it. The tool I was looking for (and found, thank God) is the CUT FACES tool. This allows you to create new lines & vertices on a poly face, allowing you to fold at "crooked" angles, and also not disturbing textures already placed.
Bruce- I'm a noob, for sure, but from all I know about the nature of polys, it doesn't seem like that would be possible- creating a single curved (or even angular) line within a single poly. You would have to create multiple polys and "approximate" your curved path with a series of straight lines through multiple polys. I think....