I live in WA where palm trees only grow where landscapers put them, and there still are not many around, and not all the kinds you see in warmer places. I know NOTHING about them. But some time last winter I was in some mall parking lot and there were these sort of short (as in short for a tree, but still taller than me) fat palm trees with small black seeds hanging from them like bunches of little grapes. I took some home and planted them in a pot on a bright windowsill and kept the soil moist, and a couple of them are now starting to come up! I am super excited, and these are one of the first things I've ever really grown from seed, so I really don't want to kill them.

Should I keep the soil moist like I've been doing? Or now that they've sprouted, should I let it get dry between watering? Will they rot if I keep it damp? Will they die if I let them dry out?

Is the pot in or surrounded with plastic and is seed/seedling container on a heating pad?

Since you've been successful germinating, it'd probably be best watering like you've been...the soil shouldn't be soggy or completely dry out. Keep moderately moist. Well-draining, but not to the point soil dries right after watering.
It's not a good idea allowing water to settle in the saucer for prolonged periods either. Discard, 15 minutes or so if excess water drips in the saucer.

If your seeds are on a heating pad, soil will dry faster. A pro using heat is speedier germination.
If inside/under plastic, slit for ventilation. When seedlings have 3-5 leaves, remove the plastic.

Do you remember anything about the mother palm other than height? For example, were leaves firm or soft/flexible?
Long or short? Distinguishable shapes?

Well, congrats, squidy...Toni