when i was at Uni my friends used to store their bikes in my outbuilding which was damp so it did lead to lots of minor surface rust.
As commented above, even if you dry the bike it will get damp due to condensation but some form of cover could help that. Maybe you could get a decent motorbike cover that’s breathable and/or has some form of felt liner.
Towel bike down after washing. Just take care around brakes to not spread fork/chain oil onto them – maybe have a small face cloth for brakes.
If really worried you could always put a little dehumidifier in the bag with the bike.
As for the shed – make sure it’s external waterproofing treatment is in good shape and there is no damp coming in through the roof, walls and floor and make sure it has some ventilation.