Remove lower unit.----Take it apart.------Evaluate all parts.------Take note of driveshaft and bushing above the pinion.-----Do compression test and post the numbers.----Pull flywheel and inspect ignition coils.
Overheating ?-----It does sound like it is running on 2 of 3 cylinders.----Post compression numbers here.----Check spark.----Check flywheel key.----Does primer bulb go hard ?-----Running with VRO in service ? ----Tested VRO oil output ?
That means you have an electrical issue in wiring harness or detective shift switch.-----The new switch is pricy .----Take control box apart and inspect.-----Your location ?
Then switch wires and report on what happens.----And there is no need to run the motor for this test.----Prop can be turned one way and not the other way in gear.
Find the green and blue wires at the back of the engine.-----One simple test you can do is switch the wires at the motor.----Report if you now have forward and no reverse.
Did you take these lower units apart before installing them.----So easy and simple to do , ----Well in my opinion it is easy.----First worked on these lower units in 1968.----That was when your motor left the factory !!
Not odd at all in my opinion.-----Are the notches on shift rods both lined up with the hole in connecter ?-----Is shift adjusted properly up top on the motor ?-----And lower units are easy to inspect internally.-----And finally you are sure it is not the rubber hub slipping inside the prop...
Some of the last production runs did finally use a temperature warning switch.----Easy to look up.----Test run with a portable tank and known good hose.