I prefer to transplant more dry than wet. If the soil is saturated and you don't have a full root system to hold everything together, the weight of the soil will cause it to sag and slough off. I like to start in peat pots for the tiny seedlings, your first transplant you just replant the peat pot w/zero damage to roots.
As far as lining the bottom of the containers I use used dryer sheets to cover the holes. Water runs right through them and they keep soil from falling out, more of a sanitation thing.