Why do I have such sloppy steering?

is it possible that's just simply bad shocks could cause this?


no

Take the car to a tire shop or local mechanic and have a front end inspection done. Many will do it for free (watch if you can it will teach you what to look for).

Some things are easy to replace yourself. A full rebuild is always nice but nothing is really wrong with simply replacing the parts that are bad when you are on a budget.