Did you leave the seedlings on the paper towel? Or did you possibly tear the roots while removing the paper towel? I would have sown the seeds in soil as soon as you removed them from the cold strat.

It's relatively hard to plant tiny, delicate seedlings without damaging their tiny roots and at that stage, they can't withstand much damage.

Your growing media looks much, much too wet. Roots can't survive in water-logged conditions, especially not new uber-delicate roots.

I agree with porkchop, your soil is overly wet. Generally, is recommended to use a good quality peat and perlite mix ... a good rule of thumb, if one can squeeze water out of the mixture when placed in hand, then it's too overly wet for emerging and young seedlings.

Did you make holes in the bottom of the container so that excess water can drain out?

Yes, I did make plenty of holes in the bottom. I did water generously, but I expected good drainage given plenty of holes....

Lately I water very lightly because I also noticed that the soil was overly wet.

What's my best course of action now?

Should I keep watering lightly, or stop watering and let the soil dry?

Or should I remove seedlings and put them in some other media?

Thanks all.

Only water when needed. You can let the material dry out a bit, just not completely. What is the mixture you're using? In addition to being too wet, It looks too dense, lightening it with perlite is a good idea. Roots need oxygen or they'll rot.

Like SeniorBalloon said, only water when needed. And it does look like you've planted in dirt, rather than a nice light potting mix. And are those plastic deli containers? please verify that water can drain easily through the holes you've made!

fwiw, I use Pro-Mix potting mixes for all of my seed growing and indoor plants. I can usually find either the All-Purpose (AP) mix or High-Porosity (HP) mix -- both work nicely, especially if I add extra perlite to the AP mix.