I have a 2002 dodge caravan with a 3.3 L engine. Am getting a little oil in the cooling system, seems more when the outside temp. is colder. Have been told it is probably the head gasket. My past experiences in head gasket failure is coolant in the oil. There is no engine oil cooler in the radiator. I do have to clean the slimy sludge out of the radiator on a regular basis. Trans. Oil level has not changed, but have to add engine oil every 2500 miles or so. Any thoughts?

You may not have water in your oil but I think you head gasket is gone. There are many ports in the head that can leak not all would mix water with the oil.

Check you trans cooler again. Here's the thing. Only one source has pressure to overcome the other. Coolant is under more pressure than crankcase oil but not more than trans line pressure. It is probably you trans fluid and just hasn't entered trans yet. Adding one quart every 3000 miles is considered normal. If you take lines off trans cooler and pressure test cooling system and it doesn't come from cooler lines, than it is definitely a head gasket problem.