If your current dampers are shot, new dampers will definitely firm your ride up. They can make a surprising difference. And I’m alluding to OEM replacements.
Make sure you replace the rubbery bits too, the seat springs, bumper stops, strut mounts, and boots if your struts have them. They’re dirt cheap to buy, and the spring seats and strut mounts make a big difference in the ride, handling, and road noise.
If you want to noticeably improve your steering response, reduce body roll, and reduce sensitivity to wind, without adversely affecting your ride, allow me to suggest you look for some upgraded sway bars. Note that reduced body roll keeps the tires in better contact with the road during turns, and this alone makes a noticeable difference.
Again, be sure you get new bushings with them (they usually come with the bar(s). New sway bar links on a ten year old car may or may not be needed, but check yours to be sure they’re not frozen or worn loose. IMHO upgraded sway bars might be the best suspension upgrade you can make for a daily driver.