Okay so I just finished Scene 12 in the TRIP and I can get to the "YOU ARE UNCONSCIOUS" screen, and I can save my game from there just fine. However, whenever I click "NEXT MISSION" the screen goes black, then it goes to another black screen that says "Somewhere in the southwest" at the bottom like a normal mission should, but then the game crashes. Is this a common issue here? Are there any known solutions?

Hello, I'd like to second this.
In my case it doesn't even show the 'somewhere in the southwest' screen, the moment I press the next mission button I get the windows crash message (I'm on Win10 and running the community patch that's linked in one of the fix threads here). Anyone got any ideas?
Edit: THE ANWSER IS THIS POST:
https://www.gog.com/forum/interstate_series/gme_crashes_after_jump_into_chasm/post8

You should visit PCGamingWiki, they mention how to fix this issue and others.

Yes. I can confirm that link posted is indeed the fix on how to get that mission to work. The good news is that you only need to temporarily rename those files for Mission 13 only. The downside of renaming those files means you can't change your difficulty level in the main menu but rather only ingame. Make sure you create backups of your original files so you don't have to uninstall or reinstall the game at a later point.