I don't know how to answer your question exactly Russell, in general Its large connection going into the starter. I used a voltage meter to see if there is power going into it when pressing the start button but there isn't or it was very little I would have to double check again. It is the 3.6 engine, is there a picture of the connections you are talking about to clean?
On the starter there are two connections. For the 3.6 one of the connections should be small quick connector that pushes into the starter solenoid the smaller of the two cylinders that make up the starter. The other should be a large gauge wire connected to a terminal post on the starter with a nut. There is also a third connection on the solenoid but that should be a large gauge wire going from the solenoid to the motor.
There should ALWAYS be full battery voltage at the large battery terminal on the starter solenoid. When the start button is pressed. Battery voltage should be applied to the smaller wire on the solenoid. Because the solenoid is case grounded through the engine block the solenoid should energize which simultaneously pushes the starter pinion gear towards the flexplate ring gear and closes contacts that allow full battery voltage through the solenoid and into the motor windings energizing the motor which is case grounded through the engine block it is bolted to. The starter motor will pull a lot of power and will reduce the system voltage. But, in general the voltage should not drop below 10.5V with a fully charged battery and a healthy engine. If the engine is seized it will drop considerably lower.
The first thing I would do is get breaker bar and socket on the crank bolt and verify that the engine is not seized. If it is not then you have an electrical issue.
Next is measure voltage at the large B+ terminal at the starter. I asked HOW you were measuring for an important reason. Meters COMPARE things. So when you measure voltage a comparison is made between the negative lead and the positive lead. If you only read 3.4V when you should have 12.6 you can have a problem on either the positive or negative/ground side of the circuit.
If you have a meter with long enough leads and clamp attachments attach the negative lead of your meter to the negative post of the battery. Touch the positive lead of your meter to the positive post of the battery. Your reading should be in the neighborhood of 12.6. If it isn't recheck your meter and battery cable connections and charge the battery as necessary. Once you achieve a reading in the right range record that number. Then move the positive lead to the B+ terminal of the starter (the large wire held on with a nut). This reading should be the same as your battery reading if so jump to the next paragraph. If not check that there is not surface corrosion at the starter terminal preventing a good meter connection. Scratching into the terminal surface with the meter tip can help to establish a good connection. If you still cannot get a good reading remove and clean the starter B+ wire and clean the wire end and terminal (with the negative cable of the battery disconnected) reconnect everything and recheck if you are still not getting the same reading you got at the battery you have a bad positive cable. If you now have a good reading move to the next step.
Have some one try to start the car while watching the meter. Due to the rapidness of your fault a good meter with a min/max feature would be helpful here. The voltage should decrease when attempting to start but should not drop below 10.5V. If it does and the engine is not seized you have a bad positive battery cable or a bad new battery (although the battery possibility is less likely). If the battery voltage stays at an appropriate level, above 10.5V move your positive lead to the bare metal area of the starter case. The reading should be 0V (no more than .1V) If it isn't you will need to troubleshoot your ground connection. If it is 0V once again have someone try to start the vehicle while watching the meter. It should stay near 0V and should not climb above .5V. If the reading goes above .5V you will need to trouble shoot your ground connection.
To test the ground connection move your positive meter lead to the engine block and repeat the previous tests and measurements. The meter should read 0V at rest and under .5V cranking. If you now get a correct reading remove and clean your starter bolts and clean them, also clean the surface of the engine block that contacts the starter case. If you did not get a correct reading move your positive lead to the negative battery cable attaching point at the rear of the right (passenger side) cylinder head. Repeat the test again. 0V at rest less than .5V cranking if you now get a correct reading remove the bolt holding the cable to the block and clean the cable end, the bolt, and the bolt hold and repeat. If you did not get a correct reading when first measuring the ground point at the back of the cylinder head you have a bad negative battery cable or battery (although the cable is more likely).
If the above procedure fails to locate or solve the problem. The problem is not in the base electrical portion of the starting system. At that point I will do my best to help you. But without the proper test equipment and without being there myself it will be difficult. But, if you did measure low voltage at the starter terminal I believe that is where the problem lies.
