What shape are your shocks in? The lowering springs put more strain on the shocks (and struts). If you have the original shocks then it is time for new ones. Also, look for oil leakage or a sagging side.
LoL, yes, that sounds like your problem. Normally, they say you should replace the shocks every 60k miles. I've found that more that more than 90-100k miles and the factory shocks and struts get pretty soft and should definately be replaced.
Shocks and struts won't cause sagging. They just cause the suspension to bounce. The shocks dampen the suspension so it doesn't bounce. You can do a simple test: go outside and push each corner of your car down hard as you can. When the car returns back up, it should simply rise to the resting position and stop moving. If your shocks are bad, it may rise too far and then settle to the resting position, or even bounce a few times if they are really bad!
Bottom line, replace the shocks and struts. It will literally feel like a new car.
I'd agree that an 11 year old car with 150,000 miles is long overdue for a shock replacement and that would be the first step in diagnosing this.
Shocks are dampers designed to dampen the motion of the body and suspension components. A car with bad shocks, as bhuff notes, almost always exhibits "under damped" body motions after large suspension transits (e.g. the body continues to bounce for a while...) Bad shocks can also cause the road wheel and suspension to "jitter" allowing the tire to actually come off the ground momentarily in response to road variations. This makes the car feel especially skittish when cornering on pavement that's not glass smooth.
A consistent jump to the right might point to other issues though. If the rear axle is not being located properly you might also be seeing a "bump steer" effect out back. If the axle is pivoting under the car (even a bit) in response to a large bump because a control arm bushing has pulled through or there are issues with the upper control arm bushings or mounts it can cause the rear of the car to suddenly steer one way or the other for a moment.
At this mileage I would have the whole suspension at least inspected and wear items like shocks replaced.
every mustang ive ever had hops over a bump but it doesnt bother me, i just changed my shocks and struts but i havent paid attention to if it still does it. If it bothers you these other guys are right just replace em
The solid axle and soft bushings help the Mustang's rear end hop around quite liberally. You can minimize some of this looseness by upgrading both your upper and lower control arms. A good setup for minimal investment is UPR's Elite kit at $ 219. These have urethane bushings and are made from chrome moly. This is a good choice if you enjoy the twists in the road more than 60' times. I have these on my Mach (along with UPR's full length sub frame connectors) and it feels SO MUCH better than stock. I am very confident running into a turn at twice the posted speed limit. (UPR also has some great 1/4 mile setups as well. Also check out their wishbone rear end locator - sweet!)