My general guide for derailleur trouble shooting goes like this. First disconnect the cable. Test the spring by pushing on the derailleur with your hand. If it pushes back, it's not the spring. Second pull on the cable with your hand, while you actuate the shifter with your other hand. If you can feel the shifter pull and release cable, it's not the shifter. Third, feel for friction in the cable. If it's rough or sticky, replace cable and housing.
My hunch is it's a sticky shifter. This things are cheap and the gunk up easily. To ungunk, spray aerosol solvent into the shifter. See if that frees it. If it does, spray an aerosol lubrincant in there, like Tri-flow. Wipe up mess.