Zandronum

At least visually. I noticed that my connection quality is probably to blame, but as a result I appear to skip around to opponents, as they skip around in my view as well. Is there a way to smooth player movement as far as compensation for this goes?

Try turning unlagged on in Player settings (it should be on by default). If that doesn't help, try playing in servers with a lower ping.

Unlagged has absolutely nothing to do with player movement. What you can try is lowering cl_ticsperupdate from the default 3.
But from my practice, this setting does little to help, some players whose connection is worse skip even more on lower values...
Zandronum needs improved client interpolation.
Try uncapping FPS also.