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This really sounds like a blown head gasket (perhaps something else as well...). The white smoke when starting is from coolant sitting on the cylinders after you shut it down. Hence the longer it sits, the worse the smoke when starting. Eating oil rather than leaking it is another sign.

My neighbor's car is in the shop for head gaskets right now. White smoke, overheated, has been "eating" oil and coolant for about a year now.......My suggestion is to get this addressed ASAP by a professional if you can. Waiting is not your friend if its a blown head gasket.

Just my opinion, of course, but I think this situation is on the verge of getting much worse if not addressed quickly. Borrow the coolant system pressure tester as previously mentioned. I'll bet dollars to donuts it won't hold pressure and you won't find a visible leak, either.

Mmmmmmmm----donuts...........

You may not be getting oil in you radiator from the oil jacket to the water jacket, but you may be getting coolant to your oil side. Check your oil and make sure it isn't emulsified (milky brown/frothy). Make sure the level is not overful, because that would be the oil level increasing with the coolant.

Blew a head gasket in a Chevy once and filled up the oil pan before I could get off the freeway. Catastrophic failure. Locked the motor up on that one.....

Have you noticed any white smoke from your exhaust, not the cold weather/early morning type, but the billowing stinky heavy stuff......

Just trying to help.....

It is in fact supposed to relieve pressure. You have a problem, my friend. You didn't put the thermostate in upside down, did you? Just checking.....Have you checked your fan(s) again? How about your belt(s) and or water pump? Not likely to be a coolant leak if the reservoir filled up and the cap was blowing steam, in my opinion.

Keep it simple. Take the belt off (usually simple) and rotate the water pump pulley. Does it spin like it should, or is there a binding anywhere? The pulley shaft should not wobble, either.

+1 on the fan/fan switch issue. Been there. Done that. Inexpensive and easy fix. Hopefully it'll turn out to be that easy.