An AAV is not complaint for a washer drain. And you cannot add another drain into the 2" washer drain. You need to find a way to vent this arrangement.
You need to upload more pictures, like of the main drain.
- Washer overflow must be 36"- 48" high.
- You put a washer trap and a vent pipe up at the red spot.
- You put a sink trap and a vent up at the blue spot.
- The two vents will join together higher and have a single AAV.
To verify: go to loveplumbingform.com
"So basically make a rectangle with a shared vent for both?"
Some states require 3" vent above where the two join. Check your local code.
In my city, a 2" vent is good for both washer and sink.
Again, go to loveplumbing.com and find the diagram that illustrates your scenario.
Lastly, do something to improve the flow at the corner with the 3 elbows. You can do better.
That is the easiest way to solve this.
I do not want the washer drain hose to hook over the side of the sink. It's too far away, and we have plans for the space between the washer and the sink.