I've had a Centaur 10spd shifter with one broken G spring but it shifted accurately. If the shifters are worn the clicks should feel soft, and the shifter should be really quiet. A rebuilt shifter has much more "solid" and louder clicks.

The rebuild is much easier than you'd think. I've never touched a shifter in my life but rebuilding the Centaur 10spd with new springs was really easy and the two springs cost like 5 bucks. There are plenty of great tutorials out there.

Are you sure it's not a compatibility issue? I've bought a used bike before with a Campagnolo groupset where the cassette was cobbled together from all kinds of different cogs and spacers.