Generally, Rigs include the signal chain from Input to module REV. The Output Section is not included, but is global by definition.
You find the Output Source settings in the Output Section. These are global and by output. Here you can determine, if an output just feeds an unprocessed signal or a processed dry signal or just the wet portion ("DLY/REV Wet"). This explains, why the SEND 1 feeds another signal than the other outputs. But again these settings don't change, when you load another Rig or just play guitar.
You can set up wet sounds per Rig using the Mix control in delay and reverb effects. You can dial beyond 100% and from there lower the dry signal portion until it is completely wet. So these are by Rig, but don't change by themselves.
If you would just be using headphones, I would say the headphones or cable are broken. Or the socket of the STAGE is defective. If you loose one contact out of three of the stereo headphones cable, it has exactly that effect, that you just hear the difference between the left and right signal. And that is the wet portion. The same an happen with XLR outputs and cables, if the hot wires are twisted on one side. If you then mix these signals in a mixing desk, the dry signal portion is cancelled out.
I cannot imagine any scenario software and ahrdware included, where this happens by itself and affects all outputs. So please verify your assumptions.