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You do not have to purchase the ticket with the card in order to receive the checked bag and priority boarding benefits. You simply need to be traveling, and your SkyMiles number must be on the reservation. The system will identify you as an Amex holder and you will receive these benefits regardless of the credit card you use to purchase the tickets. An agent shouldn't try to charge you for a first checked bag, but if they do, having the card in your possession should be sufficient to get that waived. Priority boarding will be reflected on your boarding pass automatically. It should say Zone 1 (which is fantastic marketing - Zone 1 is actually the 4th group of passengers to board the plane).

It does take some time for your Delta and Amex accounts to be linked after you apply for the card. A week is probably a good estimate. So assuming you get approved for the card well in advance of when you travel, you should be good. You can confirm when the accounts have been linked by logging into your SkyMiles account. Go to My Delta and in the My SkyMiles box on the left side of the page, it should show you as being eligible for Pay With Miles, and indicate which card you have. Once that appears, you know Delta is aware of the existence of your Amex account.

Not all airlines are like this - United, for example, requires you to purchase the ticket with your Chase United Mileage Plus Explorer to receive the free checked bag. But Delta does not.