New tires have much taller tread blocks, much more tread squirm.
Nearly-bald tires have next to no tread squirm.
This is probably at least part of what you are feeling.
As already mentioned by Jim, it doesn't help that your new tires are in a lower-performance category than OEM, and also don't rank particularly high in terms of "cornering stability" in the Tire Rack survey:
But you should be fine, just be gentle with steering inputs, smooove smooove smooove...