I would start by trying googling logic buffer and latency settings, or logic crackling examples for recording, and then for mixing, and try that a bit. It usually helps but each system reacts slightly differently.
Thanks for these ideas xgman, I am looking into them.
I think I may have made some progress toward solving this problem. I went in and switched most of the low end EQ to high end, and a good 85% of the problem disappeared.
This leads me to believe that somehow the low end frequencies, spread across multiple tracks, were causing distortion. I realize that David said (very confidently) that this sort of thing shouldn't be happening--and I don't know why it did--but that does seem to be the takeaway.