Easily possible. Your original springs are probably just fine; they seem to outlast the cars. An inexpensive set of dampers (the "shock" portion) will acheive your goals. People often sell the Biltstien (yellow) dampers for under $300, you just put your springs on them. There are a number of inexpensive, non-adjustable damper sets out there. Probably around $200. You will also want to inspect your bushings and other wear-item parts (tie rod ends), as they can be a big part of the problem.
With your own labor and maybe a cheap set of basic/used-but-servicable dampers and a few other suspension items, your budget should suffice.
I see the KYB GR-2 (basic, good damper) on Rockauto.com for $51/each.