Will taking a screwdriver to them make the tuners actually turn without bouncing back?
Yes. It sounds like your tuners need tightening. Tighten the screw a quarter-turn, tune up the string, and see if it holds. If not, try another quarter-turn. If you've tightened it too much and the tuner becomes hard for you to turn, loosen it again just a bit. I've met loose friction tuners that needed a 360-degree turn or more to get them working right.
Properly-adjusted friction tuners should hold tune just as well as geared tuners do. They'll need tightening once or twice a year. Sometimes loosening, too, if the humidity changes. I have a little blade/phillips screwdriver on my keychain and it goes everywhere with me. If I'm going to play a gig and I won't be able to tune between songs then I'll give the tuners an extra twist just for insurance.
If the tuners are really old and corroded or the screw is stripped, then they'll need replacing. If the wood in the head stock is damaged, that's a whole different can of worms. Know that the folks at the Magic Fluke company stand by their products and will do repairs, usually free, even if you bought it used.