Q: My car is hesitating and shifts hard - could that be a sensor or is the transmission going out?

If the transmission has over 200,000 miles, there might be a benefit to changing the filter and fluid in response to the shifting issues you are describing. Before you do that, though, have a professional carefully check the fluid level (hot and cold) because too little or too much transmission fluid will cause shifting problems. You want to rule out "wrong fluid level" before you try other fixes.

If the fluid level is OK, but it is very old and the filter has never been changed, and you are only experiencing hard shifts, I would go ahead and change it on the small chance that it will resolve the issue. If it doesn’t resolve it, you likely have issues internal to the transmission that will necessitate looking at the valve body and solenoids at minimum and, in the worst case, possibly a rebuild of the whole transmission. If you are also experiencing engine hesitation, that would be due to an ignition, fuel, air induction and/or engine management problem that would require a pretty standard diagnostic routine.