First, I would not put them in direct sun until I hardened them off. They look like you did over-water and I am glad you re-potted them. Did you clean the pots with a little bleach? Was the potting soil new?
Last ditch effort would be to get rooting hormone and cut off healthy leaves and stem. Put them in new soil and cover with plastic. Do not put them in direct sunlight otherwise you will cook them. Keep moist, not soggy.
Make sure that you are using very small pots. When they are putting out new growth and roots take plastic off. Re-pot when you see roots beginning to grow out of the bottom...better yet, slide plant out of pot and if there are a lot of white, healthy roots you can re-pot into a pot an inch or two larger. Allow to dry out, in fact, I allow them to almost wilt before I water again. Use clay pots, coleus seems to like the air they get from the porosity of the pot. Make sure that you wash your pots and use a little bleach before adding potting soil. Coleus do well on a shaded porch that gets a lot of ambient light bouncing off white walls, concrete. Slowly, get them used to direct sun...half an hour a day two or three days ten an hour a day...they do well indoors if they get bright sunlight filtered by windows.
Fertilize with a slow release fertilizer, and if you see white patches on the outside of your clay pot (salts from conditioned tap water), soak, clean your pot and re-pot with fresh potting soil.
When you water, water very, very well...then allow to dry out before you water again.
Do not put stones, gravel in the bottom of your pot...the soil has to become saturated before it starts to move into the bigger pore spaces and your plants will have less air. Just potting soil, lift the pot up off the deck, table, tray with tiles to increase the drainage.