Double click on the rig that you want to paste to so that it becomes active(headphone icon will appear on the line to the left) and then paste. It should work. Sometimes the paste function is greyed out and it takes a second right click before if becomes active.
In the case you describe, you may want to lock everything but the amp and cab so that you control what is pasted and ultimately overwritten.
I always just save the existing rig as a copy, then change the amp in the new one. That way you keep both and can tell if the new one is more or less suitable.
There are two ways to copy/paste an amp:
1. At the front panel using the COPY/PASTE buttons of Head and Rack models while the amp module is in focus.
2. In Rig Manager via right clicking at the amp module and selecting Copy/Paste from the menu.
The modules of the signal chain in the Editor area always relate to the Rig currently loaded - the one which gets displayed at the front panel and marked with a headphones icon in the Rig Manager list. So you need to load the source Rig via double click, copy the amp, load the destination Rig via double click, paste the amp. If you don't load the appropriate Rig via double click, you might copy/paste the amp of Rig A into the same Rig A and therefor don't experience any changes.
Hi, Thank you all for this clear explanation. 
Another way to change the Amplifier module in a rig would be to select it and then browse for a different one, either from rigs or from presets (in case you have a few amps you use more often).
This can be done from RM or from the unit itself.
HTH
For me, saving an amp preset and just right-clicking to load it in place seems to be much easier.
