Display capture is the worst thing you can do. This is what you get if you try that: https://cdn.discordapp.com/attachments/386420834002141186/555794100838727680/Display_capture.mp4
Firstly, GOG provides you with a fully functional DX wrapper, improved Diablo, better res, however you want to call it. It allows you to set the resolution of Diablo to ANY size you want. Which means, the simplest thing is to use the DX wrapper on your classic Diablo! Set resolution to your desktop's resolution in DX wrapper, and use Game capture in OBS. The steps to do it have been mentioned multiple times on these forums.
Secondly, you can capture any window using Window capture. For Windows, even fullscreen games are a window. If you decide to play the classic version for some reason, without the wrapper, use Window capture and set it to Diablo. When you got that set up, you have two options:
Either you set your Base resolution to 640x480 in OBS, because that's the resolution of Diablo, then you're good to go. https://cdn.discordapp.com/attachments/386420834002141186/555794135731404806/Window_capture_480p.mp4
You set your stream resolution to 640x480, but you scale your resolution to whatever resolution you want to stream in. For example, scale it into 1920x1080. https://cdn.discordapp.com/attachments/386420834002141186/555794169470386186/Window_capture_upscaled_1080p.mp4
Are you aware of a way to have the best of both worlds: A Non-Stretched Integer Fullscreen that has the perfect resolution and that does not negatively affect the graphical display online? Its either I get a stretched stream or I get a windowed one with perfect ratio but the channels are log-in panels are all in the top left corner of my screen when I enter Battle.net.