I think you underestimate Konerko's trade value.
I do.
Look at it this way.
If I was a GM of some team, and looking to trade a starting pitcher of mine, who is decent, and the White Sox offered me Paul Konerko for him, I'd say no. Unless the GM is a complete moron and doesn't look at speed, defense, etc, I wouldn't accept.
Granted, there's no way of not knowing that PK won't have a great slump again this year. He probably won't. I just don't want that slowness of his on our team.
Possibly a team looking for a DH or a 1B that hits for power, and don't mind about speed, then we might be able to get something useful out of them.
In a perfect world, I'd get rid of Frank, move Lee to first, DH Konerko and put the LF and CF spots up for grabs. While Carlos isn't a bad left fielder anymore, we could definitely use more speed in the lineup and moving him to the infield wouldn't be a bad move in my opinion. Then you put the CF/LF spots up for grabs in ST and let Rowand, Harris, Borchard and Reed fight for them. Of course this will never happen, but it's just what I think would be best for the Sox. I think if the Sox keep Carlos long-term, you'll see him switched to either DH or 1B.