First try the home made de-clogger. Pour 3 cups of backing 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 are right.You do have 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.. you could use a bottle cleaner with long shaft.
Wired assembly, Red showing the parts you can remove.
