I would, but it randomly started working again. I started re-creating the elements in my composition, but manually copy/psted the numbers for the various properties. Those layers worked fine though the old ones didn't. However, once I deleted *some* of the old layers, they all started working fine. I don't know why that'd happen, but I get the sense that it was a bug.
The exact workflow I did was this: I took layers from a composition where they had parents, copy/pasted them into a new one where they had no parent (I tried both with and without explicitly setting parent to "none"), got rid of all of the keyframes, edited their lengths to match the new comp length, and started trying to do position animation with them. Though I could move them normally, I got the weird alignment bugs as I described earlier.
Thank you for your help, Mo; I don't know exactly how to recreate this as those are the only steps I can think of to do over, but at least it's working now. I'll post again if it returns.