The use of "to have" in the first block and the second block is completely different. The second implies need, the first implies possession. I think it's confusing because of the ambiguity. Block 2: "I have to offer this to the gods" Block 1: "This is what I have to offer". For "This is what I have to offer," you can only differentiate based on context whether you mean "this is what I need to offer" or "this is the thing that I possess that I can offer".

@filth sorry for the confusion, I put the words in the parenthesis to indicate the direction and the level of politeness; they're not actually part of the phrase. I was still referring to the phrase as a response to an offer

I've never heard "I don't mind if I do" ever, and I was under the impression all my life that "Don't mind if I do" stood for "(Please, you,) don't mind if I do," because "I don't mind if I do" sounds rude and "Don't mind if I do" doesn't.

@user31113 it doesn't necessarily need to be unkind. It's vulgar language and familiar, but friends who like to joke around can say that as well and not get offended. It totally depends on your dynamics with your boss I guess.