Peat moss would make it wetter. Perlite would make make the soil looser, but I am not sure it is a soil issue so much as a watering one.
You are planting transplanting tiny seedlings, water when they get dry to the bottom of their root ball because that is as far as their tiny roots extend. As the plants get bigger, that zone that gets dry before you water again will get deeper and deeper. Losing several to transplant stress is not a surprise (they don't have many reserves at this point) and losing starting leaves is not an issue - they'll fall off sooner or later.
Moisture meters are notorious for lying about how much water is in the soil. A better way is to stick your finger in, or a stick/dowel/cheap chopstick into the soil. Leave the stick in for a minute and pull out, the damp sections will tell you how wet it is down there. (You can also calibrate your moisture meter the same way, but be aware that moisture meters sense salts and so can be screwed up by fertilizers.)
