Will it go into neutral, most vehicles will crank in neutral? If automatic check the neutral switch. The ground for the coil side of starter relay is through the neutral switch. You may be able to crank it at the starter relay instead of having to get at the starter motor. Use jumper wire on load side of starter relay, relay unplugged. Of course one of the pins has to be hot. Make sure tranny is out of gear, functional parking brake is set. Have a helper sitting in there with foot on the brake.
I realize that doesn't help the shifting issue? Can you check for applicable trouble codes? Some of the national brand auto stores will check codes for free. I just wondered if there might be any tranny codes?
Sometimes I'll take the shift linkage loose from tranny, then try to manually move tranny lever into park? There are safety issues, you have to be sure vehicle can't move. Safety is your responsibility.