Water comes up in other sink

Hi, we changed our sink last week and went from a 7" deep to 10" deep. We hired a handy man to install it. The next day when we started using the sink, we notice the water is not draining fast and typically comes from the other side of the sink and eventually both the sinks/bowls have water accumulating in them. If we just leave it for some time, eventually they both drain. But when I start the faucet, within a minute it starts clogging and both the sinks have water backed up.

I saw another thread where there was some discussion on Illinois rules for this, so I am attaching a picture of the piping to get expert's opinions here. As you can see in the picture, it has a garbage disposer on one side, with Dishwasher outlet attached at the top of disposer and the two sinks are connected to a "Y" that is connected to the outlet. I live in Chicago so please let me know if this meets the Illinois rules what could be the problem for clogging and what I should do next?

Appreciate your help.

Thanks, Deena

There must be a clog down stream from the "y", in the wall.

@Smooky, thanks for the response. Could anyone please suggest a DIY solution i could try to solve this.
I tried the Crystal Heat Drain opener with no use. I tried plunging by closing the other side, didn't work! Please help!

Thanks

I would take the pipes apart under the sink without the disposer and use a hand auger to clear the drain.

http://www.homedepot.com/s/Hand%20auger?NCNI-5

https://www.youtube.com/watch?v=QbS6Akn2E4k

https://www.youtube.com/watch?v=zcGmBwotdBw

Thanks @Smooky. I will try the hand auger today. I tried the Instant Power Main Line Cleaner (http://www.homedepot.com/p/100126446) overnight and that didn't help either. I hope the auger will be able to help.

As mentioned in the original post, this started immediately (very first use) after we changed the sink from a 7" deep to 10" deep. Do you think the height of the sink/disposer might cause this problem of water backing up? Not sure you can tell that from the picture I uploaded.

But the water that is backing up is pretty dirty/stinky, so may be it is a clog! Well, I am totally puzzled now, how did it go from a perfectly working sink to a clogged sink between changing the sink! I hope the handyman didn't screwup anything when repiping things!

Thanks @Smooky. Hand Auger did the trick finally. Even though there wasn't much that came back stuck in the auger, the water starting running fine. But the plunging of water on the disposer side of the sink seems to have broken my garbage disposer. Water is leaking out of the disposer. So may be it is time to replace the disposal now.

I’m glad you got your drain unstopped. If you put that Crystal Heat Drain opener in the disposer the sodium hydroxide may have eaten the casing if it was made of aluminum especially if it sat in there for several days.