I agree, this is a cultural difference. We Americans would say, "No thanks, I just ate." I seldom even say "I'm not hungry," as perhaps the implication with such a phrase is "I'm not hungry for what you you offered or will offer me." But we generally just say we just ate. I suppose this is really no different from "I am full" in Turkish.
I'm thinking purely in terms of English. When you speak to someone and they offer you food, if you have already eaten, you wouldn't say "Thanks, I'm full" like you are here in Turkish, but rather "Thanks, I'm not hungry" or "I already ate" if you want to focus on the eating rather than the hunger.
Yes, it is correct, but since we add the "um", it is not necessary to add the pronoun, so instead of saying "Ben tokum", you just say "tokum" with the pronoun and it carries the same meaning (just like Spanish, polish, arabic,...etc), and the same with the other pronouns.
Tokum (instead of ben tokum)
Toksun (instead of sen toksun)
Tok (instead of o tok)
Tokuz (instead of biz tokuz)
Toksunuz (instead of siz toksunuz)
Tok/toklar (instead of onlar tok/toklar)