Amazingrace, there are many things that can cause indoor sowing problems. First, you want to take the greenhouse lids are off of the flats after they sprout. I have much better success with the lids off once they sprout. They need air. They could be going through what they call damping off (see the link below). You did say they are drying out so you want to make sure you are watering from the bottom (the water should go in the tray and not on top of the seedlings. Next, the little seedlings need light. Do you have them under grow lights? Also, you want to make sure they aren't getting too cold. Such as sitting in a draft window.

I mainly do winter sowing (see the winter sowing forum). But I still sow my impatiens, begonias, coleus and a few others inside. I use grow lights and heat mats. Depending on the plant you may not need the heat mats after the seeds have sprouted. I keep my house fairly cold and I sow my seeds in our basement and it's pretty chilly down there so I keep my mats under the seedlings for quite a long time.

Let me know what types of seeds you have sowed. Then I may be of more help to you.

Also, you want to look into winter sowing where you start your seedlings outdoors. There's a good number of us on this forum that sow are seeds this way. I've been indoor sowing for years and I never got a petunia seedling to survive. Then I found winter sowing - I winter sow at least 100 petunias every year. And endless other annuals and perennials.

I hope this helps. I'd more than glad to answer more quesitons if you have them.

Here is a link that might be useful: Damping Off