Along with firmer sway bars, this is what I would recommend.
Sway bars control roll during cornering,
and will not do anything to prevent bottoming. On a road where one track is rougher than the other, that motion will transfer through the sway bar to the other wheel, transferring that motion to the body, as that is what sway bars do - transfer weight from one side of the vehicle to the other.
Cars on good, dry roads can benefit from heavier sway bars. Cars in the wet are set up with softer bars, allowing the car to roll a bit more, which adds bite in the wet. Off-road cars may end up with a small diameter bar or deleting sway bars entirely, so the four corners can operate entirely independently of the others. Sway bars are a tool in the suspension designers toolkit to dial in understeer/oversteer.
After several thousand miles the oil in the damping parts of your suspension will start braking down. This will cause a loss of both compression and rebound damping (the car will feel like it floats more because the tires are not making good firm contact with the ground). You do not have to wait until a shock fails and starts leaking. In many modern day sportbikes the suspension maintenance interval is about 18 thousand miles. These are systems so high end that upgrading them can easily cost over 20% of the original cost of the bike.
If we had that in cars most drivers would freak out! specially on our diesels designed for high millage. At around 100000 miles the suspension might not perform as well as it did when it was new...
Shocks do not stop functioning because of the shock oil breaking down. Shocks break down to the mechanical parts wearing out, or becoming dislodged, and leaking seals (internal or external) from the use over time. Non pressurized shocks fail in uses where rapid cycling causes the oil to start foaming.
Gas pressure shocks, like Bilstein, control foaming by keeping the shock internals under high pressure, and thus not allowing bubbles to form in the oil. Those shocks can deteriorate if any of the gas gets past the seals, in addition to the normal failure modes of shocks.
Springs support the vehicle under varying loads. The shock controls the action of the spring - i.e., preventing suspension over-travel, the spring rate of compression and extension (speed of the suspension arms) spring oscillation, etc., as well as controlling unsprung weight (i.e., tire and wheel)
Shocks
may help the OP problem, especially if the shocks are worn. If the stock springs have sagged, they would need to be replaced. But replaced with what...
-Stiffer springs at the same ride height would be one answer, but the ride will be less supple - more harsh.
-Springs the same rate but an inch or so longer would cause the car to sit higher, but the car would ride much the same.
-Softer springs with more preload can be fitted that will soften the ride somewhat, but due to the preload will have a higher rate than stock, and would solve the problem.
-And you could consider variable rate springs, too
Changing the ride height of the car has some negative effects.
-Increasing ride height will change the droop of the suspension, and will affect the camber of the wheels, possible causing additional wear on the outside of the tread. It will also raise the center of gravity causing additional body roll in the corners.
-Lowering the car will put you that much closer to the ground, reduce suspension travel, and again, change the camber of the wheels. This time, the inside edge takes a pasting. Good news, COG is lowered, causing less body roll.
You're lucky as IndegoBlueWagon has fitted various combinations and knows the results. Take heed.
frank
ps. As far as the high end shocks for sportbikes go, I can add some flavor to that discussion.
I have a set of Ohlins on my BMW k1200RS. Two shocks, one front, one rear, are about $2k. The front is only adjustable when disassembled, on my bike due to space under the fairing, and the telever suspension. The rear has two adjustments: Bounce damping and rebound damping, as well as a spring preload adjustment. Each have a separate external reservoir for fluid and gas. The service recommendation is rebuild yearly. I do it every three years, roughly 30k miles.
Ohlins are the gold standard for racing applications, but makes a HUGE difference on the street (if set up properly). Keeping the tires in contact with the ground and keeping the chassis stable is what they do. That helps during braking and turning, and in that light, I see them as a safety item.
Note, incorrectly set up shocks can be far worse than worn out stock shocks. Just because there is big money spent, and there is a nifty gold cylinder attached to the frame, does not mean the bike is better. Car is same same.
There are more sophisticated units from other vendors that have high and low speed bounce as well as rebound. Those are close to $3k for the bike.
"Aggressive?" 
