Yes, they are cognates. You are right that today "smut" and "smutty" are largely metaphorical with sexual connotations, but I would still use the words in their original way to refer to a kid having a smutty face, for example.
Thinking about it, I associate literal smut and smuttiness with the dirtiness that comes from soot, coal and charcoal. When I was a kid, in the dying days of steam railways in the UK, a smut was what you might get in your eye if you stuck your head out of the carriage window and the smoke blew in your face, or you got smutty from soot.
"Schmutzig" ist hochsprachlich, "dreckig" ist mehr süddeutsch-dialektal (die Aussprache ist in den Dialekten natürlich etwas anders - in meinem Dialekt: "dreggisch"). Im Süden sagt man eher: "Der Boden ist dreckig und muß gewischt werden", im Norden und hochsprachlich sagt man: "Der Boden ist schmutzig und muß gewischt werden".
Hmm, ich werde versuchen :) Die Übersetzung:
"Schmutzig" is standard German, "dreckig" is more south German dialectal (of course the pronunciation in the dialect is something different - in my dialect: "dreggisch") In the South one rather says: "Der Boden ist dreckig und muß gewischt werden", in the North and in standard German it's "Der Boden ist schmutzig und muß gewischt werden" (both mean "The floor is dirty and has to be cleaned")
Common sense tells us there is no direct word-by-word translation, and meanings are often spanned across several words from each side, so the relation is "many to many", not "one to one". Especially for words with very close meanings in one of the languages in question (English).
But I have no illusions that common sense would mean something today, so here is Collins: https://www.collinsdictionary.com/dictionary/english-german/muddy
Note schmutzig being the very first translation in the list.
Your smug answer about common sense doesn't belong here.
Anyway, as much as the issue is concerned, the first meaning of "muddy" is literally "covered with mud". Of course it might be used as a synonym for dirty, but, strictly speaking, it is more specific. "Muddy terrain / soil" does not automatically translate to "dirty terrain / soil.
Dirty is the commonest word as well as Schmutzig is. Any more word / translation for others to learn is welcome, but it should be indeed treated with "common sense".