First try thea home made de-clogger. Pour 3 cups of backingbaking soda diluted in 1 cup of water, then slowly pour 0.5 cup vinegar and watch it foam. You might see dirt coming up, so it is working. If you have avocado skins in there then follow your plan and open the drains. You
You are right.You, you do have an uncommon installation. In your case the right one was not screwed on properly and is leaking. Also the top left screw is crooked.
The third outlet might save you. Open it (unscrew) carefully (there might be some backup), and poke in it to unclog stuff.. youYou could use a bottle cleaner with a long shaft.
WiredWeird assembly, Redred showing the parts you can remove.
More bad news: Since it backs up in the sink, you problem is deeper. In the main line. Hope you can use the access port to clean it up.