I am not with Laminar Research, just a flight simmer from down under.
Good news and bad news. The bad news first. There is none.
If your joystick and yoke are configured properly they will not interact with each other. Others have followed this advice and the process does work.
All hardware has to be configured through X-Plane. Particularly Logitech (ex Saitek) products. No other way. Not even through your operating system. You perform the same process with other brands as well.
Firstly, make sure any configuration of the hardware through your operating system has been completely cancelled and or deleted.
Secondly, acquire a good quality "uninstaller" program and delete all files and directories you may have used and or created to install the hardware. You need to ensure there are no remnant files hidden anywhere.
Thirdly find the software that was provided with the hardware and "dump" it in the bin never to be used again.
If you have followed the instructions so far you are on the way to a successful configuration.This is particularly important with Logitech.
Fourthly have a look at the following link found at https://www.youtube.com/watch?v=hU_TuALAlJk from Michael Brown. If necessary watch it several times to understand the process. Michael owns a business that builds PCs for flight simulation. Predominantly for X-Plane. Laminar Research recommends this business as the only suitable flight simulation PC supplier/builder within the USA.
When you view the video make sure you note the process and disregard any product name used by Michael. You need to ensure you do not double allocate any button or lever or knob with or for the same instruction.
As a fifth process allocate your buttons to your requirement. If the hardware item is recognised (preinstalled within X-Plane) it already has allocated some buttons. In the video Michael changes some allocations. As Michael demonstrates, you may not allocate a process to a button for that hardware item either.
It is not a 30 second quick fix process.
As a sixth process save the configuration and give it a name. The naming process is at the bottom of the page/screen.
As a seventh process start all over again for your yoke. Give it a name as well.
As an eighth process select your aircraft and the newly configured hardware controller for that aircraft. You can use the individual controller in as many aircraft as you like, using the same name, and if configured properly it will not interact with the other controller. My understanding is that you can have both controllers connected to your PC at the same time and they will not interact.
If you find a button or lever nomination does not suit for that aircraft then you can change the configuration and then save the configuration with a new name. In doing so you may have several configurations of the same hardware item but each file is only suitable for one particular aircraft.
This is one of the new benefits within X-Plane.
Returning to your final question. The good news is you do not need to purchase another joystick.