Engrave / Note Spacing ON / Alt + arrow keys should do the job (see the image attached). Your specific case seems to be somehow too difficult for poor Dorico because of the “overlapping” items. You probably have tried to first move the notes a bit away, if they have handles, and then move the rest, if it now has a handle, and then move the notes back to their correct position?
It probably should, but at least my copy of Dorico 188.8.131.52 does not move a rest (or a note or a dynamic marking) horizontally when Engrave / Offset X is applied. Nothing happens. Offset Y does work.
Engrave / Offset X does move a lyric syllable, for example, but as stated, not a rest, note or dynamic. Is this intentional? If not, I hope it will be fixed soon.