On a used scooter with moderate mileage, it is possible that the bearings in the steering are worn or not at the proper tension.
With the scooter on the center stand and the rear wheel on the ground (elevate the front wheel; may need a friend to sit on the bike):
(1) grasp the front wheel and manipulate it looking for slack in the steering assy. If found, tighten the steering nut to the proper torque.
(2) Observe the front wheel as you move the handlebars from the center towards the side; is there any notch feel along the way ?
repeat towards the other side. The steering should have a smooth path from center to each side, if not then worn bearings.
If these two tests are ok, then I lean towards a worn front tire, especially if it has more than 5,000 miles on it.
Scoot safely,