Many, probably most, US banks do not have SWIFT codes. Instead, if a SWIFT code is required, you’ll need to send the funds through a corresponding bank. Your bank should be able to tell you who they deal with for that purpose.
The wire instructions would look something like this:
BigBankName, SWIFT Code BIGBANK
With credit to YourBankName, Acct # YourBanksAcctNumber
With further credit to YourName, Acct # YourAcctNum