For the purpose of this conversation, shocks and struts are the same thing.

More correctly, they do pretty much the same thing, but shocks are a
separate part, and struts are the shock and the spring in the same assembly.
Working from memory - which is fuzzy at times - the front of your car will
use struts while the back will get shocks. Frequently, the shock cartridge
can be removed from the strut assembly and be replaced, letting you put the
same spring back into the car.

The general purpose of the shock is to control bounce, absorb the shocks
that come into the suspension system. The spring is primarily responsible
for ride height, and when the springs get old they begin to sag, and the
ride gets soft and the suspension might bottom out easily. Replacing the
shocks will bring new stiffness back to the suspension, but if the springs
are worn, the new shocks won't last long.

You didn't say what year your car was, but my guess is that your springs are
fine, and you need new shocks. The shocks will let the car continue to
bounce after it has gone over a bump of some sort, typically you will notice
bad shocks when going over railroad tracks, but you could feel the affects
of worn shocks in many different situations, but railroad tracks jump to
mind first.

Shocks can also show up in strange wear patterns on the tires. Usually the
shocks will show up on the tires as cupping or feathering, where the edges
of the tread are cupped or feathered. Normally the cupping will show up on
both shoulders of a tire, the shoulder is where the tread wraps around to
become the sidewall. If you are having alignment problems too, then you may
see cupping or feathering only on one shoulder of a tire.

The shock absorbers are inside a McPherson strut.  Some, if not all, designs
provide for just replacing the shock absorber cartridge.  Your description
fits a worn out shock absorber...

If left alone long enough, you will also wear out all of the rubber bushings
in the front suspension.

Jim