Oh....as the light bulb goes on....now I see what you are saying. Your solenoid is ORIENTED (on a 70) with the two large posts running vertically.
Side posts...to me that says the large posts are oriented side to side.....so I will now refer to them as large and small posts.....
The large posts are wired properly per your description. One to battery positive, the other to starter. If I remember right, the one going to the battery is the bottom one, top one is to starter.
So, now....if you have +12v on both of the small posts when you have key in start position, then you have an open to ground on the small post that is meant to be connected to ground.
I concur with papyson on this.
So, one small post should have a yellow/red wire on it the other should be black. Wire colors may be hard to tell due to wire age.
That black wire is not grounded properly. Remove it from the post, and use another wire of same size to jump directly from that post to a known good ground location on the block.
Retry.