I'm certainly not wiser than Mimi, but I'll respond anyway!

I find that when I don't pay attention, I tend to sag my weight down on my hands because it's the lazy thing to do when daydreaming along. I can get almost all the weight off my hands when I tilt my pelvis back (as in tipping my tailbone down and belly up). This makes me more aware of distributing my weight more evenly between hands, feet, and seat. I also sometimes make a game of riding with just my fingertips lightly touching the bars, sort of training my core to do more to hold me up.
Perhaps this might help you.
Sounds like with all that tennis and such, your hands should be quite strong enough already. Good to avoid the elbow locking thing at all costs though- as soon as you lock your elbows, all your torso weight tends to go right onto your hands.

Good luck!