Just for the future, I'd recommend posting screenshots to Imgur and posting the image link on reddit. Makes it much easier for people to see your screenshots.

It shouldn't stretch the image if you keep the aspect ratio the same (holding shift might help here). Here's a small video of what I would do if I had your input as a source in OBS. The video quality won't look great, but it will actually allow you to do your stream/recording without the paneling.

Without using a community-documented retro upscale like something from retrotink on your PS2, you're unfortunately going to run into weird issues like this. If you want the best quality here are some videos on how to get the best picture quality from your original PS2 hardwaree and a review of the Retrotink 5x.