Why are the analog sticks on my Nintendo Switch controllers all starting to drift?

It sounds like the thumbsticks are wearing out, which is a problem common to all thumbsticks and not limited to Switch controllers. It's probably just a coincidence that you have two affected controllers.

I would try recalibrating again. If calibration made it worse that makes me think something went wrong in the calibration process. The stick is so off-center it seems like it must be a bad calibration rather than just a loose stick.

Storing the controller properly will help avoid damage to the thumbsticks. I throw my Switch Pro controllers in a backpack every day and have never seen any issues like you describe, but it can't be good for them either. Make sure the stick isn't pressing up against anything when it's in a bag or case.

Replacing the thumbsticks isn't TOO difficult but does take some time and a good screwdriver. The joy-con thumbstick modules are less than $10 on Amazon.

This issue is now known as "Joy-Con drift." Nintendo eventually addressed this in July 2019 by allowing for free repairs without regard to warranty status* in North America.

Always start with the recalibration and firmware update since that's the fastest option. They also suggest updating the system software and removing any skins on the controller. If it still doesn't work and you don't want to try repairing it yourself, head over to the Nintendo Joy-Con repair page to file a request.

*: _{The page currently states "Each Joy-Con will be evaluated by our technicians for final warranty coverage." so you may want to give them a call to confirm free repairs are still available outside of warranty if your controllers are over 12 months old.}