Greg as far as what an engaged core feels like. I can only relay how it feels to me. This assumed you are adept at ground based sports and understand a bit of kinesiology and the principles of how the body creates, stores and delivers energy in sport. I think most of us do even if we can't articulate it.
As a pitcher I understood how I created torque by anchoring my feet into the ground, resisting against it and counter rotating my core to generate torque. The arm is very passive at this stage and is along for the ride. It's the tip of the whip for lack of a better term. By generating hip speed and uncoiling that tension in the core it's transmitted up the body...UP the body from the ground up....and culminates it's move by holding that arm in place and continuing to rotate the hips until there is no way to stop the momentum. THIS is when the arm receives all that energy and the pitch is delivered. Notice I said UP. That is how you create torque from the core.
So, if you can start to think of the stroke as not beginning from the arms and working it's way down that might help you get the feeling. Doesn't matter if it's hitting a golf ball, throwing a baseball.....these actions are done from the ground up if done properly. Watch a youtube vid of a pitcher in slow motion with play/pause. Hard part is we are face down in the water, but the idea is the same. Take that feeling and try to feel the same muscles activating in the water