Could be the fan clutch isn't engaging fully as rad gets warmer, you would also notice this after engine was warmed up and you were at a long stop light, temp gauge would creep up a bit, not over heat just creep up.
After engine and rad are warmed up, turn off engine and try to spin the fan, it should move but not spin, you should feel a good amount of resistance.
And you MUST have a fan shroud, it it was left off then thats why engine would run warmer.
Could also be the water pump blades are getting smaller from electrolysis, and that's what killed the first rad.
No, a rad for an automatic is fine to use in a manual, they cool the same.