I have come to you guys to see if you can help me before I bring my car to the shop.

I have an 06 azera. When my foot is on the gas it is fine, but when i take my foot off the gas and I am slowing down the headlights and dash lights dim for a couple seconds, the engine even down shifts into first a little rough(maybe from the surge). a couple times I thought the car was gonna die.
I brought the car to Sears and had the alternator and battery checked out, they are both fine.

These problems happen in the daylight also, its just easier to notice at night with the lights on.

Please help