We added a shed roof on our barn for our camper, but the contractor didn't measure correctly to account for flashing. Before you say "sue him" I'd like to see if there's another fix. Due to capillary effect, water is dripping from in between the ledger board and the fascia on our barn. Is there a sealant or something that can be added to stop it?
Sealant is rarely the right approach in roofing. It's a temporary solution at best, and in this case it's simply inappropriate. It would be very difficult to apply to that joint, and you'd end up with a huge, ugly gob with questionable effectiveness.
I'd just slide a bit of flashing under the shingles to extend the drip edge (lapping onto the new roof). By moving the drip edge further down it's likely that you'll stop the backflow for most scenarios, though strong wind could push some back up there.
Flat stock would work, or perhaps "dormer flashing", which has an obtuse angle in it and a hem. Gently lift the shingles in a few spots and tack it down.
To help remediate wind-driven rain, you could possibly implement that flashing along with insulation strips:
