It's easy to drown seedlings. They're roots will rot in wet soil. The plant usually sits there, looking somewhat alive for a week or more, before officially croaking.
I hate to tell you "stop watering for X-days," because I don't know how long it will take to dry out, in your conditions, but I'd say at least 3 days.
BTW, overwatering is probably the most common mistake people make with seedlings. Seeds love to be soaked in water, but once they sprout a root, they want oxygen in that soil, just as much as moisture. Keep the soil moist but not wet. Put more holes in the bottoms of those cups.