The 5320.20 is only used to notify and receive approval from the ATF for moving items already in your possession.
If the dealer still has physical possession of the item and the stamp is still in transit from the ATF (I'm not even sure that really matters), I believe there is going to be a problem. My understanding is that given your current status with the new out of state license, the dealer will not be able to release the item to you. The reason being, is that the state of your driver's license needs to match the state from which the Form 1 or 4 was submitted. Someone else can chime in here with more details if that is wrong.
At this point, I think you may be faced with another 2 transfers. A Form 3 from your old dealer to a new local (in state) dealer and then another Form 4 from your new dealer to yourself. Maybe someone else can think of a way out of this situation.
Out of curiosity, why did you wait so late to contact the ATF? Surely you knew in advance that you were faced with possibility of moving out of state while Forms were pending, right?