A 650 with new tires is very nimble. I just changed the rear tire on mine, it had worn pretty flat at least 2" across the middle. I was delighted with the new tire on the back, and how easy the bike weaves on the road now (when I want it to).
But if you feel like something has changed since you got it... How many miles are on the bike? A 2008 isn't that old, I would not suspect the front forks to be leaking yet, but if you are getting oil on the fork tube I don't know what else that could be?
I would raise up the front wheel on blocks or a MC jack, check your front wheel bearings for play, and the steering head as well. If you have a torque wrench check the front and rear axle bolts (43 ft-lbs front, 67 ft-lbs rear). If you don't have a torque wrench beg borrow or steal one from Harbor Freight before you wrench on any important bolts on the bike.
If the front is tight and the wheel spins ok without the brake dragging, then block up the rear wheel, check the wheel bearings for play, and check the swing arm for play. Be careful checking the swing arm, because the shock will be extended fully, and you don't want to pull the bike sideways off the blocks or jack.
The only other thing I can think of, stand in front of the bike facing the handlebars, and hold the front tire between you knees, and pull on the grips to make sure there is nothing loose in the handlebars or triple tree. Don't go nuts trying to twist the handlebars off the bike - you are just looking for slop or free play.
Ok.. thought of one other thing: the engine on this bike spins in the opposite direction of the front tire - so the gyro action of the engine flywheel cancels out the gyro effect of the front tire. If you are riding the bike and reving the engine rpms up, that will make the bike feel more nimble - for example: if you are riding at 55 mph in 4th gear, you are spinning the motor faster than you need to, and you can feel the difference. Shift up into 5th gear at 45mph and see if the bike feels more stable.