Should a new rear shock be building oil in the shaft?
That is definitely not right. Send it back for repair.  

Similar to front shocks, a good shock will have a very light film of oil you can just feel on the surface. A wipe with a clean cloth will show no visible sign of oil on the cloth (maybe smudge of dirt).  In this case, its clear an oil seal has gone. 

Side note: It looks like you could do with lower air pressure for the riding you are doing. You have no used anywhere near the full travel of the rear suspension.