smokes bad in cold weather

Maybe it wouldn't hurt , but OP said it smokes cold -- i would think it would take a little time to build some pressure --I dunno --couple of bucks for a new cap isn't bad though.
Easy test though --start it cold without the cap -and see if it smokes or not.

 

Has the truck lost anymore coolant?

 

And a few bucks more at the Speedco for an oil sample analysis will tell the OP whether a cap is the problem before he trashes his engine...

What is this? Pretend the problem isn't a problem so it will go away maintenance?

 

Have to side on the thermostats first for a couple reasons. Its a faily cheap repair and most likely able to do yourself. And second reason is two things cause white smoke coolant in cylinder and incomplete combustion do to a cold cylinder. I garentee you look around your yards on a cold morning watching the trucks fire for the first time in 12 hours and you'll see alot of white smoke. Get your engine up to temp first then start investigating coolant issues. Just a mechanics point of view.

 

The cooling system is a closed circuit, so ZERO coolent should be getting into the cylinders. There's a difference between a little water vapor in the exhaust on a cold morning and white smoke. A big difference. Go ahead, change the caps... but get an oil sample too. Unless you want to be doing an in-frame possibly tens of thousands of miles early. What's it cost for an oil analysis? $50?? Cheap insurance that you won't pay through the nose to fix the damage caused by something as cheap to fix as an EGR cooler.