I think you may have misunderstood me, which is common. Just ask my wife.
I meant to remove those wires, and check the starter cable post with the key on and off.
So let me try it this way.
1. Disconnect starter cable from solenoid. All other wires still connected.
Does the starter cable lug have 12VDC with the key on? Should be 12VDC.
Does the starter cable lug have 12VDC with the key off? Should be 0VDC.
2. With the starter cable still off, remove all the wires from the I connector.
Does the starter cable lug have 12VDC with the key on? Should be 0VDC.
Does the starter cable lug have 12VDC with the key off? Should be 0VDC.
3. With the starter cable and the I connector wires still off, remove the wire from the S connector.
Does the starter cable lug have 12VDC with the key on? Should be 0VDC.
Does the starter cable lug have 12VDC with the key off? Should be 0VDC.
4. With everything above still disconnected,
Does the I lug have 12VDC with the key on? Should be 0VDC.
Does the I lug have 12VDC with the key off? Should be 0VDC.
5. With everything above still disconnected,
Does the S lug have 12VDC with the key on? Should be 0VDC.
Does the S lug have 12VDC with the key off? Should be 0VDC.
Let me know what the results are. Should only take a few minutes, took longer to type than it will to test.
Oh, you probably already did this, but make sure the mounting bracket (part of the solenoid) has a good clean connection to the body. That is the ground for all of this, and if it doesn't have a good ground all sorts of weird things can happen. And you said this was an engine swap, so make sure that there is at least one ground cable (braided or whatever) going from the engine block to the body/subframe. I run another starter type battery cable from the point on the block where the battery negative goes to somewhere on the subframe, wherever there is a convenient hold for a nut/bolt. But one of those braided wire ground cables is fine. I just like overkill.
Russ