Different situation now, especially sans ducks who's juicy poops can create a cess pit.
I wouldn't sweat it(haha!) too much.
Deep litter is a deep subject(geez there I go again!-sorry).
The condensation on the concrete floor is a temporary springtime situation.
A decent layer(2-3") of shavings might even keep the moist air from touching the cold concrete and condensing.
The barn lime may help absorb what moisture there is and keep the surface inhospitable to any organism growth. I don't know much about lime, except one is caustic when damp and another is not....don't use the caustic one.
Here's some stuff about lime from my notes.
http://www.walterreeves.com/landscaping/lime-different-kinds/
https://www.backyardchickens.com/threads/agricultural-lime-vs-hydrated-lime.1218000/
Hydrated lime, slake lime, or "burn lime"- This is pure white. Hydrated lime is very caustic, so the bag will have a warning on it. This is the kind of lime you use for white wash . This lime will burn you and your livestock. Do not use it on the floor. Do not breath it.
Agricultural lime, "ag lime", "garden lime", "barn lime" or dolomite- This lime is gray and can be used to spread on the floor of your barn. This is safe for you and your livestock. It will not burn.