I was in a very similar situation with a 2006 F250 6.0. In great working condition, it was worth around $15k. It started blowing the white smoke though, likely head gasket issues, and wasn't inclined to pull the cab and starting tearing down the engine. I was under the impression the repair was going to cost around $6k. So do I try to sell it for $8K-$9K or do I put $6k into a $15k truck?
While thinking about it, decided to resurface my asphalt driveway, so I parked it on the street, and the next morning it was gone. Pretty much the perfect solution. The issue being, with a Chevy, that solution is harder to replicate.
Maybe see what it's worth, if the repairs are 50% or more of the cost of the vehicle, take it to auction?