Ah. I’m pretty sure I say “cot” and “caught” exactly the same.
If I try to imitate the way “dog” sounds when pronounced by, for example, some southerners, the closest I can come up with is something like “dah-ugh” (without a clear break between the vowel sounds). If I’m laying it on really thick, I might try “dah-oog.” Is it something like that?
ETA: Shrek’s accent is quite distinct from Mike Myers’ normal pronunciation.
ETA: I’m surprised by the suggestion that there’s a difference between the (presumably first) o in “cotton” and the o in “moth”. You’d think King of the Hill would have taught me that.