I doubt they need to be bone dry before using them. I generally only shake them out really good and dry off a bit with a paper towel too. I do make sure I wash my hands well w/ soap and water before handling the sterile nebs and only use paper towels (no sense reintroducing more germs after carefully sterilizing!)
I think for many the issue w/ drying is more about the concerns over germs growing on nebs kept constantly wet, rather than whether using neb pieces that are a bit damp is ok. With the lid closed, the 'wet' environment in the Avent is supposed to remain sterile for up to 6hrs, at which point it runs again, re-sterilizing the environment, and a sterile environment doesn't breed germs. (I'm not sure if the Nuk makes a similar claim; I haven't looked into it.) That being the case, I still prefer to run my Avent an extra cycle just prior to use... both to be sure things are completely clean and for how much 'drier' they come out.
I think for many the issue w/ drying is more about the concerns over germs growing on nebs kept constantly wet, rather than whether using neb pieces that are a bit damp is ok. With the lid closed, the 'wet' environment in the Avent is supposed to remain sterile for up to 6hrs, at which point it runs again, re-sterilizing the environment, and a sterile environment doesn't breed germs. (I'm not sure if the Nuk makes a similar claim; I haven't looked into it.) That being the case, I still prefer to run my Avent an extra cycle just prior to use... both to be sure things are completely clean and for how much 'drier' they come out.