Starter spins but does not engage

Just so I understand, when you hook the battery up, the starter starts spinning with no key input? Meaning you are not turning the key and it spins but doesn't engage the flexplate to spin the engine?

If this is the case, we need to suspect the ignition switch is stuck to the start position or the relay has failed applying power all the time. However, the starter gear may be stripped as well because if the starter is spinning it should be engaging the engine.

Try swapping the relay and seeing if that makes a difference. If it is not coming on any more you still may need to replace the starter but first things first.