This is more appropriate for the Cruise Air board, here:
https://boards.cruisecritic.com/forum/77-cruise-air/
It sounds like you have booked a "codeshare". You booked through Delta, but the flight is actually operated by ITA. This means you are subject to ITA's rules, not Delta's. ITA may have rules on paying extra for early seat selection...I don't know.
Usually, you need the ITA booking number (a 6 character identifier for your reservation) (NOT Delta's, but ITA...you may have call Delta, or dig around in the Delta booking and/or website to find it). Once have it, go to the ITA website and follow directions there on finding out your flight status, using the ITA record number and your last name.