The only components involved in this scenario are the starter, solenoid, battery and the 3 cables (battery ground, battery to ebox and ebox to starter).
As mentioned in the above posts, go through each of these components and check for damage, corrosion, or general condition. Make sure all connections are clean and tight.
Some common culprits are;
Worn or seized brushes in the starter,
Seized front bearing in the starter,
Seized bendix
Corroded connections on cabling
Corroded Aftermarket cables made with copper wire instead of tinned copper wire.
Battery itself starting to fail.
Having said all that, the common culprits mentioned above tend to manifest in an engine that is slow to crank or a battery that appears to always be discharged. This is due to the fact that they usually restrict power available to the starter via high resistance. The scenario you are describing sounds like the opposite, too much load. This to me sounds more like a short or heavy amp draw above what the solenoid is rated for.
I think you are looking for something that is physically restricting the starter from turning like a bad bendix, seized or partially siezed engine, hydrolock, something stuck in the pump etc.
I could be wrong but it's worth checking out and simple enough to do. Just pull the plugs and turn the engine over by hand.
