Now, do you think the cases being SAAMI -.004 is affecting my accuracy given my chamber dimensions?
K3, I would say yes given my personal experience with my AR and other rifles. I've found that the best accuracy I've ever achieved in any of my rifles is when I use a case three, four or five times only neck-sizing as I go. As I mentioned, it may take a number of firings to conform to the chamber. With my bolt action rifles I don't bump the shoulder back unless it's absolutely necessary i.e. the round is getting hard to chamber. With my AR, I simply neck size every case and then bump the shoulder back 0.002 inches (from SAAMI + 0.001 inches which is the headspace dimension of my AR chamber) so that the rounds will feed easily and still give me great accuracy. I'm guessing that the new cases you're using are sized SAAMI - 0.004 inches so that they'll function in most rifles.
I've got one AR that takes new brass and pushes it to SAAMI +.006 or .007.
That's still within spec though. If I remember correctly, according to my Redding literature, SAAMI + 0.010 inches is considered to be within spec.