The car is a beauty. On the carb issues… I saw its an autolite, is it mechanical choke?
Stumbling when you mash the go pedal? Dont mash the go pedal! 
Is the trans a C4? C6? I dont THINK the 1-2-3 Cougars still had FMX transmissions, 1-2-3 experts, chime in!
Anyhoooo… check the vacuum line for the kick down/shift solenoid. I’m sure others will have other ideas for you too. Congrats on the work done to date!
Your car should have a FMX transmission since it is a 2bbl. There is a vaccum line going to the trans. on those that is known to get worn and leak vaccum.I would suggest you check it at the intake and on the transmission at the modulator valve. Have you checked the intake gasket for a leak?
A couple of things:
Double check that you installed the correct part number thermostat for the 351c and no the 351w. This is a frequent mistake new Cougar owners make and the W thermostat doesn’t work correctly in the 351c because of a bypass valve and brass plate in the engine. Often when the engines are rebuilt, the brass t-stat plate is tossed by the machinist. And often when people are servicing these engines, people toss them because “I’ve never seen anything like that before.” And they wear or degrade over the years too. WCCC sells a reproduction t-stat brass bypass plate that lets you use the more easily sourced Windsor engine t-stat.
Anyway, double check your part number. Or post it here and I can tell you if it’s correct, if you’re unsure.
As for the hesitation, there are a couple of things that cause hesitation. Vacuum leaks, bad accelerator pump on the carb, and an improperly set choke.
Start by double checking the choke (easiest first) to ensure it’s set properly. Then check for vacuum leaks by starting the car and as it’s idling put your hand over the carb horn to seal off the intake path from the atmosphere. If there’s a leak, it’ll pull air through there and keep running.
To check the accelerator pump, look down the throat of the carb to ensure it does a squirty squirt when you work the throttle. (with the engine off)
It should have an FMX transmission. But a car this old could have a replacement unit as any of the three available automatics in that period or even an AOD.
Quick test to see if it’s an FMX or whatever is to count the pan bolts on the underside.
The FMX will have an aluminum bolt on bel housing and tail housing, with the middle being cast iron. Count the pan bolts if you are unsure and post that information and I can tell you what you have.
When I had problems with hesitation it was because of one of two things, or both. When I took the top off the carb to check the float levels, I found tons of little grains of rust sitting in the bottom of the bowls. I got all of it out with an eyedropper, or almost all of it anyway. Then I checked the float levels and one of them was off. After adjusting that float and getting all that rust out of there, the hesitation was gone.
As a matter of course, I replaced the fuel tank, sender, and lines, because the tank and/or lines were obviously the source of the rust grains. Very glad to have done that. I already have a rebuild kit for the carb sitting on my table and should be taking care of that in the coming winter weeks.
If you haven’t already checked the float level that might not be a bad idea. Can’t hurt, anyway.
Good luck!