Having dealt with the grooves thing myself, the best thing I ever did was changing out my front tire for one without a centerline groove (Pilot GT in my case). The worst was when driving with stock tires on a road at lower speeds that had just been rotomilled , in preparation for new pavement. That 'tracking with the groove feeling' made me very uncomfortable, and I haven't had it since going to a crosshatched pattern without a centerline groove.
The stock Connie tires aren't very good, and are holding your bike back. Keep in mind that Kawasaki went with the lowest bidder for those tires, and it shows. There are many better options out there, that will be better suited to whatever riding style you prefer (sport, touring/high mileage, etc.). I will never put the 'stock' tires on a Connie ever again, as the bike handles MUCH better with some of the other choices which you may see recommended elsewhere on the forum here. From what I've read here over the years, almost all members here will concur with the assessment of ditching the stockers for better options when the time comes to get new tires.
I currently have a Pilot GT front and an Avon AM42 Venom rear in 'wing' size as I'm a road warrior type, not a sporty type. The Pilot GTs are discontinued, which is why I currently have a Venom on the rear, and have several interesting options for my next tire change. I have 45K+ miles on the bike currently, and my current tires should be good for at least another 5-8K miles.
I'm STILL pissed off with Michelin discontinuing the GT's. For those of us that didn't mind the heavier steering, they were great high mileage tires that handled well; not quite as grippy as some stickier Avon combos that some Connie owners use here, but definitely much better than the stock tires, and definitely much longer lasting tires. But I digress...
Definitely check the bearings as mentioned above, of course. Also, keep in mind that the front wheel bearings aren't that expensive, btw, should you feel the inclination to replace them yourself.
BTW,
www.murphskits.com is a good source for Connie stuff, and he happily stocks bearings for the Connie as well as a lot of other stuff.