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As far as I understand it, the usecase for nutzap is slightly different than offline receive. For nutzap, it's more a commitment from the sender that the receiver is a specific pubkey. It also makes sure that only the receiver can spend the ecash. However, there are no "offline checks" for nutzaps, and it is also not really necessary. Zaps are essentially tips, and you don't care about it so much if someone sends you invalid payments. I think you will just know at the moment you swap the ecash how much you have actually received. If you're trying to receive a payment while offline, as a merchant or a vending machine, you have to make sure the payment is valid. because you exchange goods for the payment, and if it's invalid you got rugged.

I'm not an expert on nutzaps, i haven't used them myself. DLEQ proofs ar a system to validate if the signatures that have been issued on the secrets are actually from the mint. You need to know that, otherwise it could be some random signature and you'd only realize onve you try to swap