Also: 1. Bump your rear suspension preload up all the way. If the back end sags, it takes too much weight off the front wheel and the front wheel "hunts". 2. Bump up your rear tire pressure. Same deal. 3. Bump *down* your front tire pressure to see if that helps (but not below the minimum in the KLR manual). If it actually hurts, or makes the steering "squishy", try to bump *up* the tire pressure to the pressure listed on the sidewall, and see if that helps. 4. Pump a little air (10psi or so) into your front forks, see if that makes it more stable. If so, then your front springs have sagged and you need to put in new front springs. (Kawasaki builds their front springs out of compressed oatmeal, they only last a couple years before sagging). Note that riding offroad with air in the forks will likely blow the fork seals out, so this is a diagnostic, not something you want to leave in the forks.
If you had knobby tires I would suggest switching to a street-oriented tire like the Sirac, but I see you already did that :-). But I'm pretty confident that by making sure your head bearing is tight (should have just a little drag in it when you move the bars) and getting your springing set up right, you can get your KLR working quite well. I ride my KLR at 70mph (115 km/h) every day to commute to work with absolutely no problem at all. However, to do that I had to significantly change my suspension setup -- I now have a heavier spring in the back, heavier fork springs in the front, and this got my KLR absolutely stable all the way to its top speed where before, the front end tried to "hunt" or wobble at 70mph or above.