Do you have appropriate tyres? What about alignment?

What brand of tyres do you use? Are you still using the original tyres?

When you say shake/slip, are you referring to wheel spin in the wet?

If you have the AWD there is no spare tire.  Toyotas answer was to make the tires run flats.  My 2018 AWD Sienna factory run flats are like hockey pucks in winter.  They start to be less grabby once it starts getting closer to O Celsius.  Buy a set of third party alloy rims.  The toyota ones are super heavy.  Then get season appropriate tires.  All seasons should be called no season.  Those run flats are heavy as well.  My set of winter tires on aftermarket alloys are 5 pounds lighter.  While this may sound like it wouldn't be a big deal the unsprung weight savings instantly had a noticable ride improvement.   It helps the tires stay in contact with the road surface rudcing wheel spin.  If you are going to leave the city throw one full size spare in the hatch. Donuts are garbage anyway.  Inside the city if you get a flat call CAA/AAA.  So to sum up reduce the unsprung weight by getting a set of aftermarket wheels and get season appropriate tires.  I bet this would make a huge difference.  One caveat if you go out of town and have one of the full sized spares and use it you can seriously damage the drivetrain if the circumfrance of the tire is much different.  You can compare when buying your new tires by looking up the tire revolutions per mile.  I kept my winter tires within 2 or 3 revolutions per mile which was a fraction of 1% difference.  I'm not a mechanic so confirm everything I told you before you commit with your trusted shop of choice.