2000 Dodge Intrepid Starter problem

This 2000 Intrepid died while driving down the road. When it died, some electrical smoke blew inside the vehicle. The starter had stopped working and had some visible melted & charred debris on it that smelled like the smoke that came inside the car. So I installed a new starter. It now only makes a whirring noise (gear on the starter is spinning but the gear is not being thrown forward to engage the flywheel). I checked the small wire (that goes to the starter) for voltage and it does get 12 volts when the key is turned to the start position. Swapped the starter- relay with another identical one in the fuse block and got the same result (whirring noise only). Could this problem be caused by the ignition switch? Should I check for voltage on the large wire when the key is turned to the start position?