Something here is not making sense. Lets back up a bit...
Starters do not turn by themselves. You have to provide power to them, and a *lot* of power at that.
So don't look for any mechanical reason for the starter to be "engaged all the time". It *is* engaged (through a starter clutch) all the time. What determines whether your starter turns or not is whether or not the battery is connected to the starter motor.
Once again, starter motor needs a *lot* of power to turn. That means they need a lot of current at 12 volts - like around 30-40 amps. That also means the wires feeding the starter motor are big thick conductors. This makes your job easier since you can ignore any smaller gauge wires for this problem. You don't need a meter for this one.
Look at your battery (+) terminal. Follow the big heavy thick red wire to your starter solenoid. This will be a screw post connection on the solenoid. Follow the big thick wire on the *other* screw post on the solenoid to the starter motor input post (also a screw terminal).
That solenoid is what connects/disconnects the battery from your starter motor. The way it works is to short those two screw terminals together (hooking the battery directly to the starter motor), or disconnecting the two terminals (disconnecting any power to the starter).
You say you completely removed the starter solenoid and the starter still turns. This is not possible, since removing the solenoid will disconnect all power to starter at it's source. It would be like if you told me you removed the battery and the starter still turns
. So I suspect that you are perhaps misidentifying something else as the starter solenoid.
Another thing that often happens is that the lugs to the 2 bolt terminals of the solenoid get twisted and short to one another, thus bypassing the solenoid. You should be able to see this visually.
And the solenoid could indeed be bad if it is shorted across the two terminals inside. But removing the solenoid would stop the starter from turning. This is another reason why I think you are misidentifying the solenoid.