As long as your fogginess isn't from condensation, it should be an easy fix.
You can usually find out by searching online where the release button is for the stem for your particular movement. I usually clean mine's with eyeglass cleaning spray and tissue. I use a loupe to check for lint and streaks. I'll use tape for any remaining specks or residue.
I've found that the fogging most likely will recur but it'll happen slower over longer periods of time.
Not sure what causes it; my guess is it's probably gases released from something curing like plastic spacers, rubber seals, glue, etc, sorta like the "new car smell" process. JMHO.
I've done this to at least 4 or 5 watches - all Invictas.
The only issue I had was with my last cleaning on an Invicta vintage; when I reinstall the stem, the date wheel won't advance. Everything else works.