Yes, it means "feeling or showing caution about possible dangers or problems."
Wary and beware come from ware, a middle English verb and adjective.
The Wary-Wary word for south is habagatnon. See link for more.
Wary, alert.
The deer was wary of the predator following her.
Desirous, wary, and envious.
ware, wary
Be wary of that soot on the hearth.
'Warily' is an adverb, derived from the adjective 'wary'. 'Wary' is a word used to describe someone who is for example very watchful, is on guard, or 'cautious'.
The homophone for "wary" is "weary."
This versatile chess master is a wily tactician; be wary.
The lifeguard was very wary when he saw the little kids go in the water. Mom's hands were trembling terribly as she warily watched her child be taken off.
He was a very aggressive character, so people were always wary of him.