From the description it sounds like it could be missing a gear as it shifts. So, instead of going from first into second it's going from first into third. That would give you harsh engagement between gears. You might try changing the fluid in it but with 200,000 on it you are more likely looking at a transmission rebuild if the rest of the car warrants that.
This would give you an idea of the transmission service:
https://www.2carpros.com/articles/how-to-service-an-automatic-transmission
Oct 3, 2022 at 2:15 PM