I think I figured it out. It was a clock source problem. I noticed--quite by chance--that the problem went away when I opened up a new Logic template. I was going to record the sound and post it here, if possible.
That made me think there was some conflict with the Axe Fx and Logic, so I started fiddling with settings. I noticed the noise went away when I changed my project sample rate back to 44.1 Khz (from 48), but my guitar now sounded thin, and there was a lot of noise beneath the main signal. Because it seemed to have everything to do with the sample rate, I started looking for places where I could make those adjustments.
Anyway, long story short, after messing around with the front panel of my MOTU 828, I wondered if my clock source had somehow changed. The MOTU directed me back to my computer to make any changes. In my MOTU audio/midi settings, clock source had changed to "internal". I put it back on SPDIF. Seems like the problem is solved.
I hope.