I believe "to walk" is okay, but I want to know if there are other verbs (non-derogatory) more acceptable for people that use walking sticks.
The first word that springs to my mind is hobble, which means:
hobble (verb) walk in an awkward way, typically because of pain from an injury: he was hobbling around on crutches.
The word doesn't necessarily have negative connotations; here is an excerpt where it was used in a 2011 news story (emphasis added):
In another conciliatory note, Rep. John Dingell (D-Mich.), the “dean of the House” and longest-serving House member in history, swore Boehner in as speaker at 2:15 p.m. Wednesday. A moment later, Dingell hobbled with his cane to the Speaker’s desk to congratulate Boehner.