With two vacs and booster you are good to go. We got second booster about 10 days before departure as it was our decision, no one in Italy EVER looked at the back of the card.
We could never successfully upload any forms to Delta prior to leaving for Italy 2 1/2 weeks ago. Took all paperwork (CDC, Locator formsm, Passport) in a folder to check-in at Delta, was done in 2 minutes. Boarding pass in hand. You may be able to upload 72 hours prior to departure, likely will get another message from them. We did from DL but still could not access Fly Ready app.
Only wrinkle is that DL announced in ATL that EVERYONE would have to get a paper boarding pass, have their temperature taken at the gate, and a notation on that boarding pass prior to being allowed to board. Had not heard that one before.