PS I don't have a place to plant them outside so I think they will be staying houseplants (for the foreseeable future anyway) so that's how I'm treating them, and that's the state in which I want to know how to care for them. (ffff, weird sentence, sorry.) I have never had indoor trees so I reeaally don't know what I'm doing, on several levels.
I don't want to put them on the porch until they are bigger, either. I think the tallest one is only about an inch.

Hi Squidy. Too bad you don't know the type of palm. There are desert and tropical palms. Many have swollen trunks.

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