I rarely fly Delta, so can't opine on their process/procedure.

However, with American, there's usually two lines - one for Priority passengers (not just First class) and one for everyone else, by group numbers. However, it can be quite a mess, especially at the onset of general boarding. If you can make your way over to the Priority lane, I believe they'll take you at any time during the boarding process. Sometimes I've seen them rope off the Priority lane, but I don't think that means it's closed. I think it's just to keep general boarding in one lane. If you come up to the Priority lane and indicate that you have a First Class seat, they'll usually open it up for you.