“Unhappily” applies to the wandering, “Unhappy” to the wanderer’s state of mind, and “Unfortunately” to his situation, I think.
I’m not sure I understand what it means to wander unhappily. Doesn’t it simply mean that the wanderer is in an unhappy state of mind? I do agree that the phrasing in English has a somewhat different meaning, but what is actually going on seems to be the same action and the same state of mind.
It’s a fine distinction, but a real one.
“Unhappy” denotes the nature of the wanderer’s mind, “unhappily” the nature of his wanderings. Not much difference in this instance between the two, but there are some.
My big dictionary gives “malheureusement” as “unfortunately”, but as an adverb in the sense of someone’s mood, “d’un air malheureux” (or “d’un ton” if referring to something spoken.
“Malheureux” can be the adjective for both “unhappy” and “unfortunate”. But for the sense of personal unhappiness, you could perhaps use “mécontent” for the adjective or “avec mécontentement” for the adverb.
“malheureux” is both a noun and an adjective, applied to someone who is in a state of unhappiness, or suffering bad luck.
“malheureusement” describe circumstances that are not optimal, as in “malheureusement il n’a pas survécu” = “unfortunately, he didn’t survived”
J’errais tristement par les rues froides et sombres.
This allows you to translate an English adverb with a French adverb, but one that sounds more intense to me than the original.
J’errais sans joie par les rues froides et sombres.
This is closer to the intensity of the original, but you have substituted the adverb for a noun phrase (“without joy”).
In other words, it depends on whether you prefer sticking as closely as possible to the syntax or to the connotation.
Does “sadly” mean “unhappily”?
To me, sadly is more intense than unhappily.