How do I eliminate the cold drafts coming from my air conditioning vents during winter?
The air in your duct work is cold because it is running through the attic which is not heated. There will be some air flow through the ducts because they are not a "sealed container" they are designed to move the air in your home. 

As for the DIY project if your covers are metal I would suggest using a magnetic tape or adhesive sheets to attach them. Or they make covers:

For the ceiling registers you will want to make sure the magnets are strong enough to hold material tightly.