I will first leave this article, taken from WWDC 2014, which discusses Apple's attitude on using hamburger menus: http://blog.manbolo.com/2014/06/30/apple-on-hamburger-menus
Here is another article that actually discusses user engagement in a real application that switched to a drawer (aka: hamburger menu) and then promptly switched back when they realized what they had done: http://thenextweb.com/dd/2014/04/08/ux-designers-side-drawer-navigation-costing-half-user-engagement/
If a drawer is still the "right choice" for your team, here are a few points that come up based on your description.
You are hiding all the content, not just sliding it over. Traditionally a drawer will slide out but still show the application "off to the side". Such as Mailbox:
What you are doing is changing the view. The fact that your drawer is sliding in from the side, the top, the bottom, or wherever, does not matter. Once it is "open" it consumes the entire screen and is, therefor, an independent screen; it's not a slide out menu anymore.
You might as well have a button labeled "Menu" that reloads the screen to the show menu, and a button that says "Back" (or similar) to return to your content screen. It does the same thing and is actually much clearer.
Your swapping a non-directional action for a directional action. A hamburger menu has no direction. Given the below, which was does it act?
It doesn't. Once the menu is hidden you are swapping it out for a directional icon:
Why is there direction to one action, but not another? If your button "points" to imply the direction it will close, it should point to imply how it will open too.
By not totally hiding the contents (see the point above) the hamburger icon is still visible and is still a target for the user to hit to close it. "I hit it to open the menu, perhaps I can hit it to close it... sure enough, it worked!".
The directional arrow makes no sense anyway, since you've removed all context that this is a drawer. The user doesn't know there is content that has "slide off to the side" to make any sort of direction meaningful. A button that reads "Close" (or "Back") is meaningful.
If an arrow must be used -- your content slid in from the right, and closes to the left. The arrow points left.
How do you best solve the problem? In order of usability:
- Don't use a hamburger menu (aka: drawer)
- If you do, don't cover up all the content.
- If you do, don't pretend it is a drawer; label the buttons accordingly ("Menu" and "Close").
- If icons are a must have, it points left.