For how long has tis bike been out of use? Normally I suggest a good cleaning of the drivetrain and then LUBING. It all gets stuck without it and some bikes has not seen lube in years. Clean the rear der and especially the little plastic jockeywheels. If you need to take it apart do only one at the time and lube before you put it back together again.

Put one drope on each of the small "joints" on both the front and rear der`s. Twist and move stuff so the lube gets worked in.

Lube the wires: Gear down if possible front and rear. Work on one cable at the time: Unhook the housing and slide it away so you can lube the wire under the housing. Do this to all the foru cables and whipe off the exess oil. Only if this does not work would I start buying parts.

If you are not into the stuff maybe you should not open the shifter. Just spray some thin lube in there and see if you can get it mowing.