Veggie seedlings

I planted veggie seeds in cardboard egg cartons last week. Most are peeking up now, but not all. I keep them on a tray which I push under a patio chair at night to protect them (until I have my grow space). That means they don't get direct sunlight now, and I'm concerned about that.

The sun here is very strong, and I'm worried that putting them in the sun will result in them drying out completely or burning up. But I also don't want them to get too leggy before I plant them in the beds.

Would it work to put them inside my garden fence where they would be shaded in the afternoon but get direct sun through the morning? Is there something I can do to help keep them from drying out in this windy and hot location? The egg cartons are on a plastic tray which I could flood with water once or twice a day (whenever it looks like it needs it). I can also use plastic shopping bags as a sort of greenhouse, which I've done before. But I don't want to smother them or baby them so they don't harden.

It's a big difference growing veggies here and back in Wisconsin where seedlings we're EAGER to soak up all the sun they could!

Thanks all.

Did you sort it out?  

One thing I've found with egg cartons is the seedlings have to be transplanted out pretty quickly or they get stunted.  I notice this when I use the cardboard rolls from toilet paper too, but not necessarily when I make similarly-sized tubes from rolled up newspaper.  

I`m starting seeds here in Brazil and have exactly the same issues. My seedlings are on a card table in the front yard (they were supposed to be in the back yard, but recent wild weather made me bring them closer where I can protect them at night, we`ve been having freak hail). I rig up shade cloth that cuts about half the sun, kind of like what you might find in a garden center. Before I was able to buy it, I used some old gunny-sack kind of bags I had hanging around. In the past I`ve used sheer scarves tied up like a tent fly. It cuts the worst of the sun, otherwise you end up having to water 4 times a day.
I have found after a few years that the normal seed starting flats or egg cartons just arent deep enough, the seedlings just fry. i made my own sprouting boxes out of scrap wood, but I have also used whatever kind of container that has at least the depth of a shoe box. I`ve noticed a real difference in seedling survival.