Is it a transmission problem when the 2023 model S experiences a dragging sensation similar to hard braking when you floor the accelerator, and then suddenly shoots forward?

The RPM might be increasing too quickly, causing continuous gear shifts. Try not to slam the accelerator. It could also be that the front wheels are slipping. You might want to have someone check if the tires are slipping. If they are, the TCS will engage to help you regain traction before speeding off again.

When you floor the accelerator, the engine speed increases rapidly, causing the transmission to downshift quickly to adapt to the high RPM for rapid acceleration. This quick downshift often results in what's called a dragging or jerking sensation. Almost all fuel cars experience this during sudden acceleration, except vehicles with a CVT transmission, which feel different!