The delay spillover settings prevents the clearing of the delay lines when switching presets. If you think of the Axe-Fx as a virtual collection of effects it's as though you simply took your delay pedal and instantly moved it around in the chain and changed it's parameters. If you're going to a preset that doesn't have the delay block then there will be no spillover.
You can have spillover by simply putting the delay block in the new patch. If you don't want delay in the new patch then set the input gain to the delay block to zero. If you do this you'll get spillover but no new delay. If you come from a patch with no delay block to one with delay, you'll get no spillover as delays lines are cleared if they are not used in a preset.
If you want delay to spillover from the last patch, but you want a different delay in the current patch, use two delay blocks in your new patch. The first should match the parameters of the other patch, except that the input will be muted. This will act as the spillover patch, as long as the number of the delay block 1/2 matches the last patch. Now you can add a second delay block of whatever type you desire for the new patch and use it normally.