Just replace the cable, but it could also be a ground switch. However, realize that trash-canned lawn mowers can have more than one problem.

Mus: Follow RB's advice . You could hang the cable from the garage wall and spray some penetrating / lubricating oil down the cable sheathing and leave overnight , tedious job while attempting to free up the cable the next day . It could possibly re-seize later . Sears is good with replacement parts sourcing . Bring in the cable and Model Number / Serial Number . As for the Muffler some were machined surface without flange gasket necessity. Is there visual evidence of blowby ? You could make your own gasket if you feel it's warranted or again check your Sears Repair facility for availability :)