Todd,

I think that the stuff with oil you put on may make anything other than sand/strip/replace feasible.

For a temp fix try using mineral spirits to clean the dull areas and let dry. An application of water based polyurethane should bring back the luster (it's available in flat, satin and glossy sheen). It needs to dry without traffic for a while. It will not last that long and may cause issues with the adjoining areas, assuming they were likely finished with a non-water based version. Shellac may be a good alternative as it is quick drying as well. Will probably need many coats.

+10 on Desperado's suggestion or floor replacement is likely the best permanent fix.