Mark,
Yes, solenoid is back right.
You can use a jumper cable (large gauge!) to jumper around the solenoid, be prepared for sparks! My solenoid has a button on the top that you can push to turn the engine over! I believe it is a Lucas part that is still available, about $25. Ebay or others......to get the stock style. A Ford style will also work!
I would start with checking voltages, using a common ground. ie good clean chassis point. Then moving only the positive lead of your meter, check the connection at the battery terminal post, then the lead of the cable that connects to the post, then the end of the cable at the solenoid. All should read within a couple of tenths of a volt. ie if you start with 12.5, should have about 12.3 or so by the time you hit the end of the cables at the solenoid.
Then move the negative lead of your meter to the negative post of the battery, check the same three points on the positive cable. If these readings all vary by a vast amount, you have a problem with the ground connection to the chassis.
On my car, I found that I had absolutely no negative connection to my chassis what-so-ever! Once I installed a lead from my ZF to the rear chassis cross brace (center of the brace, you may find a post) ALL of my electrical problems went away!
If all connections and measurements turn out OK, the last piece is the cable from the other "not hot at rest" side of the solenoid to the starter. If the direct jumping of the solenoid with the jumper cables resulted in no change, then this cable or either cable end could be bad. Check for corrosion buildup at the end of the crimped on connectors.
Finally would be the starter. If you can, you can also use the jumper cable to test the starter by going directly from the starter connection post to either the hot post of the solenoid or the battery,.....again, there will be sparks, so perhaps sacrificial pieces or short cables/wires could be connected to your various hot points that you can connect to with the jumper cables and when you make sparks you won't be cooking up "goose parts"!
If all this jumping out fails, then I would pull the starter out and look at it. A high mileage Ford starter will wear the bushings and allow the center armature to ground out against the outer windings, causing symptoms similar to yours. A bench test will also fail here.....if this is the case. Otherwise the starter should spin freely! If the brushes and armature contacts are still OK, you can replace the worn bushings fairly easily and then bench test the starter for free spin condition.
I just found starter bushings, both ends, available at RockAuto.com. $2 for both! You also may find them locally! Keep in mind that your starter nose is a "manual transmission" nose and an automatic trans version will not work! Will fit, but will grind...! So if you simply find you want to replace your starter due to very worn parts, ask for a 1968-1970 manual trans Mustang, Cougar, Fairlane etc.... The stock starter should spin over the healthiest of motors very easily. The 68 302-4V was the highest compression motor Ford build except for the 71 Boss 351 and the 69 Boss 302. Same basic starters!
Lemmeno what you find!
Steve