Replanting sweet basil, am I doing this right?

Those don't even have a set of true leaves yet. Do not replant them until they have two sets of true leaves. They're too young and fragile to manipulate yet.

And yes, they do look pretty wet.

I don't see a problem, so I can't say you are doing anything wrong. You may want to cut back on the watering, they look a little wet. Past they're doing fine to me.

I agree with Marduk they do look a little bit wet at the moment. Other than adding a wee bit compost to the cells and making sure they're firmly in contact with the compost, they look pretty good to me :0)

So far good answers. The only thing I can add is to sprinkle a thin layer of peat on the soil to help prevent damping off disease.