I'm not sure if I understand correctly what you mean. But for atomic swaps you also need two accounts (on each blockchain, or with each token if you're using two different tokens on the same blockchain). Atomic swaps are a connection between two transactions of different tokens where none of both can be done separately, but both are technically separate.
The scripts used in transactions could be used here, all happening in one network but with two tokens that could have different values.
Probably however you could open, in a single transaction, a lot of NFT channels with some OP_RETURN magic in a few UTXOs.
Here is an idea and I don't know how possible it is but maybe we could use the new Taproot scripts here too. In Taproot the spending script can have different branches where only one has to be revealed when spending on chain. One of these branches could contain information regarding the Omni tokens (the amount being transferred, the signature,etc.) which removes the need for OP_RETURN and creation of channels from such transactions and the involving complications.