You're heading right on tires. I would just say unless you really know what you want and why, I'd stick with stock sizes.
The squiminess is most likely steering head bearings. If the bike has 10K or more and they've never been changed that raises the likelihood quitre a bit.
Put it on the centerstand, have someone weight the back so the front goes up in the air, or jack it up. Turn the handlebars thru left to right "lock to lock". When you reach the dead ahead position be alert to what you are feeling in the bearings. If it feels like it has a "detent" in the bearing, like it wants to stay pointing straight ahead, then they need to be replaced. Not just serviced, replaced. Stock stuff is OK but will wear again. If you've been here long enough you've read many of us talking about upgrading to tapered roller bearings. Once installed they are lifetime service and not too expensive. about $50 i think for a set.
If you have the "detent" in your steering and I'll bet money you do, it will make it hunt. When you lean gently one way or the other, the detent will try to draw steering back to dead center and you have to keep correcting. Same thing happens on a road with a crown as you have to steer against the crown to go straight, while the bearing is fighting you.