[ossiblue]

I would use either grout or caulk, keeping in mind that caulk will make it difficult to remove the drain ring should you need to in the future.

The concrete base being wet is normal and the shower pan is designed for the purpose of allowing moisture to seep through the cement and flow to the drain via the waterproof membrane, underneath. You'll notice there are weep holes in the ABS neck of the drain (or there should be) that allows water to seep into the drain through/beneath the cement. Excessive moisture on the cement can, over time, cause it to break down so repairing the grout between the tiles and especially the drain ring is needed. The point is, you don't need to seal it to a waterproof condition.