LAN games are basically HTH/8-player, and that never supported skins afaik, but the offline bit is odd to be sure.
Try re-downloading one of the skins (just one) from the AH downloads links, and go back in and try offline again.
I usually used to let AH download skins in the background while online, and didn't have too many problems.
When in doubt, go delete your cache directory and rebuild it.