When you first encounter Plague Spores, they will either be flying towards your colony or picked up by your expedition vessel.
For this achievement to work, you need to recycle your Hospital, if you have built one, and have no active Plasma Turrets.
In the case you didn’t get the achievements during the campaign playthrough, you can still get it fairly easy.
Start a new campaign mission and set the difficulty to “Insane”. It doesn’t matter which mission you choose, but you need to produce a lot of nanites. Kelori Strand worked best for me.
Build an expedition vessel as fast as you’re able and don’t put any extra workers in the building. As soon as the vessel is idle, go to the expedition map and look for a “plague spore” mission. Send your vessel to destroy it. The vessel will return having failed its mission and with sick crew members. Remember not to built a hospital, fast forward the time to 8x Speed and the achievements will pop shortly thereafter.