Thanks for the suggestions!
I've got a Sennheiser gaming headset that wasn't too cheap so don't want to change headsets, so yeah a USB one isn't an option. Neither would be a bluetooth headset, though I guess I could get a 2nd cheaper work headset, but that's just more desk clutter.
It's got 2 separate cables, one each for the headphones and mic.
So that Nobsound device wouldn't work, but it mentions on there they have another model that would, this one is what I'm looking for:
https://www.amazon.ca/Nobsound-Audio.../dp/B07GVFGB6N
The only downside of that is it has a pot in the middle which could introduce noise if it's not a well made device, and a number of reviews were complaining about the noise.
EDIT: Maybe I should just get some parts and make one. All I'd need is the jacks, a proper switch, and some wire.