IMAGE is THE SHIZZLE!
NUTSEDGE is a tuber, and you can pull that shit until you're blue in the face. It'll just come right back from the network of roots below the ground. Image doesn't work fast like Roundup does, but it's the best only thing that'll take care of nutsedge (I assume that's what you meant).
It sounds like your yard is in bad shape otherwise too though. Make sure the drainage issues are adressed and all that. The compaction may not be your only problem. Is it getting sufficient sun and water?
But it's late in the season. The cooler months are good if you're going to sod the lawn. Gives it more time to develop the roots below rgound before the leaves above ground start growing. So if you're going to re-sod, then do it in the winter. If you're going to seed, do it in early spring and get a sprinkler system.
Also depends on the grass variety...