Any thoughts on why my carrot cake layers sink in the middle?

It could have too much liquid in it or too much leavening. Instead of fooling with it, though, I'd find a new recipe unless you're terribly attached to it. I use Martha Stewart's recipe, it's basic and it's never let me down. The only things I do are replace all of the white sugar with brown and the raisins with chopped walnuts. http://www.marthastewartweddings.com/224153/carrot-cake This makes a lot of batter, but I just cut it in half for a decently sized 8" cake.

The recipe calls for a Bundt pan for a reason. Many recipes for cakes that include dense ingredients like carrots -- or fruit -- call for Bundt pans because the inherent structure of the batter when baked just doesn't work in a standard round pan. A Bundt pan is much more forgiving, due to its shape. (Look at the comments on some of the cake recipes here on Food52 that call for fresh fruit and you'll see a lot of problems where Bundt pans are not used; in flat round pans, those cakes tend to bake properly around the edges but the middle remains dense and either doesn't cook all the way through or doesn't rise.) If you must use round pans, you're much better off finding a recipe for a carrot cake that's been well tested by a reliable source specifically for round pans. ;o)
P.S. I haven't tried this particular cake, but any cake recipe with King Arthur's name on it is likely to work well: www.kingarthurflour.com/recipes/king-arthurs-carrot-cake-recipe

A few things come to mind, baking powder past its date or opening the oven door before allotted time needed.
I also recall reading this a long time ago that if your eggs are not at room temperature it would affect the baking no matter cake or cookies. Hope these help.

My guess is you are over-folding the egg whites. It's important to lift and fold, NOT STIR in the beaten egg whites. Or you could be over-beating the egg whites, so that when you fold them in they deflate.