This is a confusing picture.
Are you capturing both the desktop and the game with a monitor capture? Or do you use a different source for the capture of Payday? In the latter case you would probably just have to resize the source in your OBS preview. For example select the source that captures payday and press CTRL+F or right-click the source and select Position\Size and then "Fit to Screen". Then payday should take over the full size in your obs preview and of course also on your stream.
Use game capture, and turn Windows Aero on.
Where you have the Payday 2 Gameplay. Right click there, go into add then select game capture. Once there you need to hit refresh so that Payday 2 show up in the listing or select use hotkey, open the game and hit the hotkey that you set.
Aero basically helps prevent capture stuttering as far as I understand it. But I'm no expert on that one. I just know I've always had it enabled on my setup in windows 7. On the few occasions that I mistakenly disabled it I was getting some video stutter.
Glad you got it worked out.
Aero helps with keeping it smooth, But the big thing is with Aero off, any window you capture, will capture anything on your desktop in that area. Aero forces it so only the hooked window shows up, while hiding everything else.