I think there is a wire that runs from the starter solenoid to the fuel solenoid that opens the fuel solenoid in the start cycle. I believe there is a diode in that wire to keep power from feeding back through the wire in "Run". If that diode failed would the starter solenoid be activated in 'Run"?
Cannon

That is correct. A diode is a current flow directional device. The diode (circled in red) allows current to flow from the starter's battery cable lug thru the fuel solenoid when the starter solenoid is activated (key in "start" position). Then when the key is released to the "run" position, the diode prevents current from flowing back towards the starter motor.
Usually when diodes fail, it is because they burn out (go open). In this case the diode would appear to be shorted.

To verify this, the diode lead will have to be removed from the starter, then apply 12 volts to the "R" terminal of the key plug. Use a test light, connect the alligator clip to battery ground and probe the lead removed from the starter. If it lights, you have a shorted diode.

Okay so I disconnected the battery, energized the R terminal with a test light on the diode wire removed from the starter and it lights up.

The black barrel hickey a couple inches up from your test light probe is the diode.
So now reverse your polarities. (i.e. apply the 12 volts to the ring terminal and put the test light probe into the key plug "R". If it still lights, your diode is shorted.
You can buy the assembly, Google "959-3689 diode assembly" for about 20 bucks
https://www.google.com/search?q=959-...t=gws-wiz-serp

or

you can buy a diode for a couple bucks and replace it yourself. I understand a 1N5400 diode is a good replacement.
https://www.ebay.com/sch/i.html?_fro...A&_dcat=260820

If you replace it yourself, pay attention to the line at one end of the diode. That determines polarity and you must put it in the same way as the old one.
(i.e. end with line towards the carb.)

Okay, I'm going to have to go back out tomorrow, because I already put my toys away LOL.

So I put the test light on our and lit up the diode and the test light lights up, so it lights up doing it both ways.

There are specialty diodes that could give this indication, but in your case, no specialty, so you have zeroed your problem down to one thing.
Now you can consider how you want to fix it.

Just want to say that your diagnostics and directions are spot on in this thread. Nice work, and great job explaining it.

Thank you.

So I ended up ordering the diode assembly and a new fuse holder, so hopefully this will do it, we will see. Thank you in advance for the time you've taken to help me.