I havn't seen inside oue of those Hobarts, but the old Millers I am familliar with have a wire wound rheostat for the amperage controll and a similar linear unit in the foot pedal. There is resistance wire wound around a ceramic insulator, You can see if there is a burned off section on it. A carbon brush rides in these wires, it needs to make good contact too.
The contactor You hear clicking needs to make proper contact too. If dirt, insect nests, etc. got between the contacts while in storage, that could cause the problem You are having.