Sure sounds like the engine is sucking in something that's not supposed to be there. Tranny fluid will definitely burn white, and look like fog out the tail pipe depending on how much is being burnt. It is surprising how much moisture rides along with oil vapor in the crankcase ventilation system.

First thing I would try is getting the engine up to temperature and smoking, then remove the oil fill cap (place a rag over the hole if necessary) and see if it makes a differenece. You may have to wait a bit for any moisture in the system to burn out of the exhaust. It should be possible to remove suspect hoses and lines (plugging where necessary) to see if you can isolate where the fluid is coming from.