If you have adequate drainage, you should never have to worry about it getting too wet.

I prepare the pots with damp soil, and wet it down thoroughly once after the seeds have been sown (sink spray attachment) to make sure they are in good soil contact and that all the potting medium is settled.

After that, I really don't have to water again in my climate until later in Spring, tops are off and things are growing - can accomplish that with my hose on a mist setting.

And Duane is right, if you have a good draining potting medium and adequate drainage holes, your soil really cannot become too wet outdoors. Otherwise, with the amount of coastal rain I get, I'd never have seedlings :)

Ditto what they said. I fill containers with dry mix, soak them in the sink, top off the mix after it settles, and wet one more time. Then they drain in the sink for a few minutes or up to a few days, depending on my availability for sowing. I think I have 4 containers draining in the basement laundry sink as we speak.

Martha

I put about 2 cu ft of mix in a utility tub, wet it down with about 4-5 gallons of water and mix it really well. I then put about 3-4 cups of mix per gallon jug, tamp it down a bit, then sow. I am doing hundreds of jugs, so I don't have time or room to let a few drain at a time, so I don't add so much water that water runs out the bottom of my jugs (since I sow in my laundry room area and don't want a huge mess everywhere).

I use miracle grow potting mix, which is pretty easy to wet. (I guess things like promix are harder to get wet.)
I place six or so two liters bottles (with dry potting mix in them) in a plastic tub.
I take over to the sink and run water, slowly over the top, open bottle. Until it starts running out the bottom. Place back in tub and so on. Then I sow the seeds. Then take the tub outside, and take the bottles out the tub. Doesn't make too much of a mess that way, and I don't have to wait for them to drain.