So do you fish your seeds out of the moss before you sow them after stratification?
I pull the bags out of cold storage on the date designated by the stratification period . . . or at least I check on them if it’s been awhile since I last peeked. Occasionally, I have to act quickly because I’m surprised by a bag of already growing seedlings. Most times, everything is fine and I will leave them in the fridge until a weekend I can deal with them or until the weather is more conducive.
When I am ready, I’ll take the bag from the fridge and leave it in a warm spot in the house (70°F/21C). I check on them every few days. I’m primarily looking for the appearance of the radicles from the seeds but, I’m also looking for any mold growth now that things are warm.
Once I see the radicle emerging, I set up a pot with the best type of soil for what I’m trying to accomplish. My past experiences with a particular species might influence my soil choice also. The soil goes into a pot and I pre-wet it. I clean my hands, sterilize tweezers, whatever pointed tool(s) I’m using to make planting holes, and a scissors to trim away sphagnum (if it doesn’t separate from the root readily).
Once I see things “popping”, I will carefully dump the contents of the bag into a large clean container. If I’m late to the party, and I see cotyledons, stems, and true leaves, I will carefully cut the bag open to prevent damaging the seedlings during removal from the bag. I only pick out the individual seeds that have already shown signs of germination. A sign of germination could be as little as a cracked seed coat revealing a swollen seed that’s ready to rock. Everything else goes back in the bag until the late bloomers decide to show up. Some species will germinate sporadically over three years. I usually get tired of waiting after two years and toss the seed that’s left into a grow bed.

Here is a bag of Cornelian Cherry (Cornus Mas) seed that I’ve been picking through for the past couple of weeks. You may note from the dates that this is a bag I cycled back into the fridge because Cornelian Cherry may require a couple of cycles of warm/cold/warm/cold stratification.

I have to get on top of these this weekend because the radicles are really extending.

Here is a pot of Medlar seedlings (Mespilus germanica). I just planted these last weekend. Most have broken free from their seed coats and some are even putting out true leaves.
Please take into consideration that the way I do it isn’t necessarily the best way, it just works for me based on my climate, space, and resources.
I wish you success whatever methods you use.