Can my drywall act as an air barrier?
Drywall is mostly composed of compressed gypsum and a paper for structural integrity.  For all intents and purposes air will not flow through it if installed correctly, however the importance of being air tight is to control moisture and condensation from forming inside of your walls.  This is where the vapor barrier comes into play.

A vapor barrier to be most effective should be airtight from the outside.  This can be done using appropriate tape and materials that you can get from your local hardware store.  User decker has a great explanation on this here.

https://diy.stackexchange.com/a/25277/6086

Being in a cold climate your inside of the house will be a lot warmer than the outside for most of the year, so condensation will form on the insulation facing the outside. This moisture can result in mold.

If you are certain that there is not correctly installed vapor barrier behind your insulation, then one option is to tear down the drywall to where you can do this.

A better option might be to look at possibly rewrapping the outside of the house in a new wrap and seaming it to help create the air tight situation that you want.  You mentioned that the outside looks in rough shape so perhaps tearing off the siding was in the plans anyway?

ADDITIONAL NOTE

You may see that there exists a special moisture resistant drywall product typically called "green board".  This will not help you with moisture inside the wall because the moisture resistant properties of the drywall have to do with the green paper on the front face of the drywall.  The board itself is still composed of gypsum and doesn't have any special moisture resistance by itself without the front paper.  It is intended for installation in bathrooms where it will withstand high moisture environments inside the home.