Managed to get Delta to switch times on my booking, but maybe because I had a weird situation...
DH booked early on Glitch Day and got confirmed immediately, but the hours kind of sucked (Memorial Day weekend Departing JFK 12 Friday, Arriving LAX 3 pm, and then Departing LAX 4 pm Monday, Arriving 9 pm) and would essentially only leave us 3 days in LA over Shabbat. I was booking at the same time, but due to overlapping dates the system cancelled my ticket automatically as a dup.
I tried calling a few times to have the times of the confirmed ticket changed but each time was told that there were absolutely no changes allowed. I also wanted to cancel my toddler's ticket since we decided not to take her but they would not refund it. So I took a diff route. I told them about the duplicate booking and they said if I managed to get the contract number from Priceline they would honor the second booking and cancel the first. This was awesome for us not only because of the better times (D 7 am, A 10 am, and D 11 pm, A 7;45 am) but also because I had not booked DD on my booking.
After staying on hold with Delta for eternity while they verified a bunch of inconsistent information with Priceline, they actually switched the times of the confirmed booking and refunded DD's ticket.
As an aside, Delta had changed the times on the return flight after my booking, so the Priceline itinerary actually showed different times than what Delta was seeing on their end. They refused to change us to a different flight that had the original times.
I guess my long-winded point is, if you have a strange situation it still doesn't hurt to try!