Your complaint is that it's unstable, and you want it to turn and corner better?
Either your description of the problem is off, or you're asking for the impossible. A bike that's stable and solid in a straight line will be sluggish around corners, and a bike that's nimble around corners will be twitchy on the straightaways. You can't have a bike that's both solid and nimble around corners simultaneously.
There'are plenty of different degrees of compromises though, so if you don't like the way yours is handling you can always measure some frame geometry specifics and then go shopping for another bike more to your liking.
Or you can try swapping out the fork first, forks to contribute a lot to how the bike will handle.
Tires are a so-so thing, and unless you're doing something really strange at the moment I don't think changing the tires will do that much to alleviate your problem. But it's always worth trying a couple of different tire pressures.
On a road bike, you shouldn't be able to run too low pressures, as that would most likely cause snakebite flats.
But you might be running too high pressure, which can make the wheels kinda bouncy.