I too have a Jenn Air (parent company is Maytag) side by side that is about 7 years old and I'll be the first to tell you it is a pirce of s#*! and I would NEVER buy one again. Yes I did buy this one because I thought it would be a quality appliance, boy was I wrong.
I too had a problem with the the freezer coil icing up soild. The problem was, there is a computer controlled counter that actually counts how many times the freezer door is opened, it uses the freezer door light switch. The frequency of the defrost cycle is predicated upon the door opening. Good idea if only it would work. My computer stopped functioning and when this happens the cooling coil does not defrost at all. Eventually it will freeze soild. How long this takes depends upon humidity levels, frequency of door opening among other things. Mine would go for about 3 to 5 days before freezing. Once this computer panel was changed, the freezing stopped. This cost about $250 and the computer panel is located in the top of the refrigerator.
I have had many other minor problems with this unit including the doors not wanting to close very well, interior shelving and drawers comming apart, freezer drain line plugging up, outside of doors sweating in summer months causing the finish to rust (stainless steel finish), water in door button sticking on and water running everywhere. Like I said, this thing is a piece of s#*! and I'm done with Maytag products for good.
I hope you have better luck.
Jay