So, assuming the sentence makes more sense in the context of a conversation ... in English, this might mean something like stepping from rock to rock across a stream, or stepping carefully through an area with obstacles ... something like that.
Does the Hungarian version also imply something like this?
I don't think it would work in Hungarian. "lép" means taking one step or a certain number of steps, it doesn't fit in a sentence with "sétál". If someone is stepping carefully for a while, you should rather say "lépked" or "lépeget" (frequentative form). "Nem sétálok, hanem lépegetek." sounds better but still quite far-fetched.
Maybe the Hungarian marching song might help with the concept of stepping :)
Aki nem lép egyszerré, nem kap rétest estére...mert a rétes igen (or nagyon) jó, katonának az való bal... bal... bal- jóbb- bal bal...bal... bal- jóbb- bal, stb ...
There are different versions of this, and it has at least one more verse. Meaning- whoever doesn't step together won't get any rétes, (flakey pastry filled with good things: almás- apple, mákos- poppyseed, vanilla sugar and maybe some lemon, túros- cottage cheese, which can be sós or édes- salty, or sweet, with vanilla sugar, and of course, meggyes- sour cherry. You definitely do not want to miss out on having your rétes...) because the rétes are very good for a soldier -left left, left right left, etc.
It is genuinely good to march along to :)