I tried -giant_move and -ginormous_move at someone else's suggestion with -check_flip, but without any other suggested revisions, just to see if they made a difference. The alignment looked the same--that is, just as bad.
If it helps, I noticed in the bad alignment cases, even when I did giant and ginormous move, the following warning messages popped up:

Center of mass shifts (-cmass) are turned off, but would be large
-at least one is more than 20% of search range
Cmass xyz shifts = [numbers varied depending on the dataset]
Shift search range is +/- = [numbers varied]
Center of mass shifts turned off, but would be TERRIBLY large!
At least 1 is more than 50% of search range
Cmass xyz shifts = [numbers varied depending on the dataset]
Shift search range is +/- = [numbers varied]
Cmass was turned off, but might have been needed :(
Please check your results – PLEASE PLEASE PLEASE
No output dataset will be calculated
I also saw multiple messages about considering running “3dWarp -deoblique” if I am “performing spatial transformations on an oblique dset or viewing/combining it with volumes of differing obliquity.” This message appeared several times, listing several datasets—it would mention one set a few times, another set at different times, if that makes sense. Then when I opened the data in AFNI, I get a message saying the underlay/overlay pair of datasets have oblique angle differences of x degrees (x varies; I've seen 6.898816 and 14.807095) and again suggesting I run 3dWarp -deoblique on this and other oblique datasets in the same session. I don't know where I would put the 3dWarp -deoblique command, though.
Thanks to everyone who offered suggestions so far.