It's hard to know without seeing them but as long as there is sufficient drainage and no standing water in it as you've said there isn't, you should be fine. Mine are all pretty wet right now too. You want to keep the soil moist and not let it dry out, although you don't want it to be a muddy puddle either! Lol* I haven't been winter sowing as long as some of the others here, but in my experience so far unles the temps get up to about 60 degrees I don't open the covers at all. If you have cut a few holes in the tops or used bottles/jugs and left the lids off they should get plenty of air circulation, and the idea is to generate a little greenhouse so conditions will be warmer in the containers than outside.

That's my two cents anyhow! ;) Good luck!

Welcome to winter sowing. I find that once the weather warms up and the soil thaws, it seems that it is very moist, but this soon changes. What about condensation? If that is reasonable the soil is not too moist. A hint I got from this forum many years ago was to ensure good drainage by punching a few holes on the side of the containers (check pots in the store and you will see they do it). Once you are sure you have provided proper drainage, just let Mother Nature do her thing. I know, it is rather unnerving the first time since you don't know what to expect, but I don't think the seeds are in the containers long enough to rot, and seeds will germinate in water. So just be patient, and things will be O.K. Good luck!!

They will be fine, even floating in water. Several times over the past 8 years, I've actually had my containers floating around in a foot of water from flooding rains.
Right beside the floating containers where the ducks flapping away.

In the end, the water drained from the yard, the ducks went away, the containers went back down on the ground, and those seeds just did their thing and germinated just fine.
Even the seedlings in the containers that had germinated already where fine.

Just last year, at this time, 450 containers where having themselves a grand old time in flooding rains, 14 inches in 2 weeks to be exact and they did just fine.

Take a deep breath, and sit back and relax. your container will be fine.

Fran

Thanks everyone. Good point that seeds will germinate in water.

Fran, did you happen to get a picture of your pots and ducks floating together? That must have been just precious.