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Is this an automatic or manual?
Well The relay you are looking for is located on the driver side, under hood. See image
Here is a wiring diagram of the Starting system if this will help. It almost sounds like you may have a bad ignition switch. You may want to unplug the signal wire to the starter so you can run some tests.
The big double looking relay is for a manual transmission. I would start by unpluging the trigger wire at the starter, so it won't spin. Not the 12mm nut, the plug. Do you have a test light? or a Voltmeter? I can walk you through some checks if you have both of those tools.
It does sound like you may have burnt the starter. & Yes you can push the car while in gear, it's just tougher.
Now with the relay unpluged check voltages at the relay connector. Key off test, then Key on test. This will tell us where the voltage is coming from to keep that starter going. Tracing electric problems is like being a detective.
Tell me wire colors also, please.
This is sounding like it may need an ingition switch.