So, it seems in OpenGL there are modes the system can be in, with names like GL_PROJECTION and GL_MODELVIEW. And each one is a "stack" of matrix transforms, through which the various objects you then create are converted on its way, each frame, into being displayed.
If you change the camera, you really want it to be in GL_PROJECTION mode. What is more, if you don't call gl_LoadIdentity() before you actually change the camera, then it seems your camera's position is modified relative to its previous position instead of from the arbitrary origin of the coordinate system.
It seems like a simple mistake to make, and to fix, but my Google searches were not an appropriately-shaped key to unlock that knowledge. Most of the tutorials I had seen had covered using OpenGL for 2D, but not for scrolling around, at least not in the way I was wanting to use it. Because I didn't perform this necessary step I was getting strange behavior, like the camera skittering off when I tried to move it.
I had been partly misled by a tutorial that said you could set the projection system to gl_Ortho once, at setup, and leave it alone. The thing is, the code ended setting the mode to MODELVIEW. If you then change the camera after that, well, you get weird behavior. Instead, I needed to change back to PROJECTION, reload the Identity matrix, and move the camera from there -- changing it back to MODELVIEW afterward.