You state it starts running hot while sitting still, then cools down as you start to travel. Have you blown the radiator out? By that, I mean have you taken an air hose and thoroughly cleaned the radiator? It's best to blow from the engine side to push the dirt and bugs back out the front of the radiator. After using the air, you can follow up with water from the back side to push even more crap out. Spraying air or water into the front side of the radiator should be avoided. That pushes the dirt/bugs deeper into the radiator. Now, this is what I was always told and what I do to my vehicles. I've never had a problem with the radiator overheating due to it being clogged.
I'll ASSUME you have the 2.4 motor, as it's the only one with 2 T-Stats, but what YEAR and mileage. What is overheating, High gage reading or boiling out coolant? Did you bleed air pockets out when you refilled the coolant? Did you try a new rad. cap?
WHEN is sometimes doesn't want to crank? When it's hot or a first start attempt?