You don't give the miles, but it appears from the poor ride and the side-hopping that you need new struts and shocks.
The smoothest and also the least expensive items are Monroe SensaTrac.
Sensen are inexpensive but hard, and may have quality problems -- I had to get rid of mine.
Sachs are original equipment, but expensive, and mine rode poorly at 32,000 miles.
KYB are a good balance of ride, handling and cost -- what I finally got.
Koni and Bilstein are expensive. Their street models ride alright and handle well.
If your tires are old, they can also be contributing to a poor ride. The tires on the car I bought were originals with 32,000 miles and plenty of tread. But the car had a wiggle at low speed and a vibration at road speeds. New tires got rid of that. With new tires, struts and shocks, the car became smooth and pleasant to drive.