Is it really a matter of, I’m gonna have to buy a whole bunch more HDDs to copy everything over to, encrypt, copy it all back? What’s the best (Linux) software for a mass copy task like that to be sure it works correctly?

Yes. If budget isn't an issue then I would even recommend a whole new rig for the process, and here's why: there are several Linux distros now that have the option, upon install, to do whole disk encryption where you have to enter your passphrase to get the computer to do anything at all. This is probably the safest, easiest solution available to you. You now have a brand new computer that's never not been encrypted to copy stuff to. RAM and drives will hurt, but I think most modern processors have cryptographic instructions built in, so it may not take a large processor at all.

The downsides are that it's a bad time from an economic standpoint to be doing stuff like that, and that you're eating a large amount of technical stuff all at the same time. If you don't get it right then you might lose all your stuff.

For me it would mostly be my desktop and my nas.

I’m gonna have to buy a whole bunch more HDDs

Depending on when you started storing data and how long it's been since you touched your current solution, it might be just a few larger hard drives: https://www.backblaze.com/blog/backblaze-drive-stats-for-2025/

Go ahead and add up how much storage you'll need and decide if you're doing any kind of RAID. I recommend at least a mirrored RAID.

Give us a hint about what RAID you have on the NAS? How much storage used/available?