Fresh 727 first start up

After putting in the required amount of trans fluid I always put the rear end on jacks and run it through the gears filling the trans as I go. If you have put fluid in until it shows "full" on the dip stick you should be able to run it through the gears with the rear end up on jacks without causing any damage. But yes, clean out the cooler first but remember, that will require more fluid before start up. I never put a new trans in gear with the rear wheels on the ground.