The way you describe the breakaway sounds like you have set the car up for a relatively grippy tire, but have not given it said tire. Also if the behavior is worse in the rain, that would corroborate that you are experiencing a tire grip issue.
14k really isn't very stiff on these cars. Ek mentioned that the motion ration on these cars is between 50 and 60%, that's true. With a 17x9" +35 wheel, it's right at 0.5, or 50%. On the other IS I'm helping to set up, we are currently at 20k springs up front and 18k springs out back, with the same TRD sway bars you have. Even with that setup, we are getting more body roll than we want. If you go by the calculation alone, you'd need something like 32k springs up front to get the bounce frequency of a GT style race car.
So, 14k is totally reasonable, even with your sway bars. It depends on your tires and alignment setting as well, and your driving style and experience. Also keep in mind, the stiffer and "racier" you go, the more likely it will behave badly in the rain. The good news is, disconnecting the sway bars like you were thinking of will go a long way to alleviate that problem.
With that open differential, lifting the rear inside tire will always be an issue. The large sway bars make it worse. The BEST solution is a clutch-type LSD, but of course those are pricey. An extremely common setup in the Miata world is to beef up the front bar, leave the rear stock, and tune to taste with spring rates. That allows the differential to work reasonably well while maintaining a good balance.
One more thing, the harsh bouncyness you are feeling could totally be in the shocks and not in the spring rates. The first, easiest, and freest thing to do is turn down the shock adjustments by like, 50%, and just go for a drive and see how it feels. On most single adjustable shocks (assuming that's what you have), the adjuster affects mostly rebound, and you can accidentally dial way too much in just by nature of the damping curves. That will give the car a really harsh ride, and make you bounce out of your seat. Give that a shot.