Will I be able to get my second boarding pass without having to go through security again?

In the US you can get boarding passes for all domestic flights either:

• The classic way - at the airline check-in counter
• at any of the self service kiosks once you enter the airport
• get it printed when you online check-in on the airline website
• on your phone in the airline's mobile app

So feel free to choose any option that suits your travel needs. BUT what would be a headache for you is your checked in luggage. I am 99% certain that your checked in luggage will not be transferred from the AA plane to the Delta plane. Instead you might have to collect your luggage from the baggage claim area and drop it off on the next airline's collection counter. If this does not seem to be a hassle for you then you are good to go, otherwise it's worth a try to call AA customer service and ask them about this particular scenario.

Now let's come to the more interesting part. If you are traveling just with a personal item + a hand carry you can USUALLY move on to the transit area once you arrive and wait for the next plane near the designated gate. BUT you would need to have the next boarding pass with you i.e. either check-in online and print it or use the airline app.