You should have 120 vac to only one side of the contact when the relay is de-energized. When energized you should also have 120 vac to both sides of that contact, one side going to the outside black switch.
When the inside switch is activated you should have 12 vdc to the coil of the relay to energize that relay and send 120 vac to the switch.
If you are getting 120 vac to one side of the contact but not to both sides of the contact when you energize the relay, then the relay is bad, or you are not actually getting 12 vdc to the relay to energize it, or the relay coil is not properly connected to ground.
Since you replaced the relay it doesn’t seem likely the relay is bad. Check to see that you are getting 12 vdc to the coil when the inside switch is actuated. If you are and the relay doesn’t energize, then check the ground wire to the coil.