Recently had my tire guy replace only my front tires on my 03 Ford Explorer. Drove it on the highway and it felt and sounded like the transmission and drive train were hitting each other. Pissed. Having him now replace the rears as well. Hopefully they will now be happy together. Fronts were new rears were worn and different tires. Will the new tires reset everything I screwed up?
Hopefully. Not enough info to guess. If there was a large enough difference in the fronts & rears and 4WD was engaged (if you have 4x4) than perhaps the transfer case was struggling. But I can’t tell from here.
Let us know how you make out.
I am surprised that the tire shop agreed to put the new tires on the front. The latest guidelines are to put the new tires on the rear if only doing two tires no matter the drive train. I think Mountainbike is correct and all the tires should match.
Sssshhhhhhhhhhhh … don’t tell them! It’s so good to hear of a shop willing to put two new front tires on. We don’t want to spoil it! 
The part that really concerns me is that the OP says rear tires were worn and the front and rears were not matching. You would think with liability problems the tire shop might have insisted four new tires of the same type.
I can’t answer your question because I don’t know the details of your 4WD or AWD system, but I am wondering how many miles you drove on the mismatched tires.
Regardless, your car is safer in low-traction conditions and in emergency maneuvers with matching tires and matching tread depths, especially because the bad tires were on the rear, so there’s no sense in getting mad about the situation.
I’d be surprised if this doesn’t improve things straight-away.
I am not a Ford expert, but from what I just read, 2003 was the year that the Explorer went from the Control Trac 4WD to Ford’s intelligent AWD. Either of those two systems should be able to handle mismatched tires IF it is working correctly. I suspect there are problems here other than tires, but if it works OK with matched tires, I would keep driving and not worry about it.