Most greenhouses have fans to circulate the air to prevent damping off as well as other problems. I don't know if this is your problem but they shouldn't go limp.

Thank you, I have no fans, it is a small, homemade greenhouse...I will try circulating some air. Thanks again.

I'm having the same issue...

Oh, dear, peat pots. You must take some small scissors and snip out all but one seedling, then get them into larger slightly pots (3 to 4 inch diameter). Then start giving them more light. Dill doesn't hold well when young, so it's important to get things moving along. Next time try little plastic cups filled with seed starting mix. Then you would be able to "prick out" and transplant extra seedlings. Those peat pellet pots are pleasing to gardeners, but not so to many plants.