Car shakes when downshifting and stalls sometimes

I got this car used and I am trying to diagnose some intermittent transmission issues.

When I am in a high gear and I start breaking, the car shakes a lot before each downshift.

If I brake the car too quickly it stalls when it stops, and I have to put in park and start again.

Sometimes even after starting the car again it shuts off when I put in Drive.

Should I start by changing transmission fluid and filter as first step? If so, would I be able to save the fluid in case I have to change something else later on?