Answer: Unfortunately, peat moss is problematic if used alone. It sounds like the seedlings may have been too moist and are damping off and the seeds are turning moldy as a result. Here is a run down on seed starting to avoid damping off.
Start out with fresh clean soilless mix (usually peat moss plus perlite and vermiculite or similar) that is just barely moist, like a well wrung out sponge. Plant your seeds lightly and cover the pot with plastic wrap to maintain humidity. Place the pot in a bright location out of direct sun and watch for sprouting. Open it as soon as they start to sprout and put them immediately into bright light.
Make sure there is some air circulation where you keep the seedlings as stagnant air can encourage fungal growth. Next, make sure the plants are thinned enough to allow for ample light and air to filter through them and to keep them from having to compete with each other for nutrients and moisture. Also, do not over fertilize them. They do not need fertilizer until they have several sets of true leaves.
Keep the foliage dry; try watering by dribbling water gently and slowly onto the soil rather than spraying them -- wet foliage will also encourage fungal growth.
Finally, make sure all of your tools and equipment are clean and that you are using a relatively sterile potting mix. Other causes for damping off can be too high or low a temperature or too little light -- many gardeners find it necessary to use supplemental lights for seedlings. An ordinary shop light fixture using one "warm" and one "cool" bulb is usually adequate for seedlings if left on about 15 hours a day and if the leaves are kept within about an inch of the lights.
Last of all, you might have luck watering them with chamomile tea when you see the fungus appear. At this point, I am not sure if moving the seeds to a drier environment will help or if they have rotted in place, but it might be worth a try.
I'm sorry you have had such rotten luck; gardening is full of "experiments" and we tend to learn a lot the hard way. It does get better with experience.
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