Is the water valve continuing to leak after I shut it off normal?
As keshlam points out, old valves don't always seal completely; try to be quick. Also, even if think you've drained the system, it may drip for quite some time. Pipe-fitting doesn't mind a little water; apply dope and screw it on. 

Note- The T&amp;P valve is to be piped to near floor level, with no threads left exposed to be perhaps capped by someone; cut them off.