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.
Sink drains come in prefixed design. Nothing can be done wrongly there, but you are right. In your case the right one was not screwed on properly and is leaking.
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.
