Water backup on new double vanity/both sides
I might be wrong, but I'm "Picturing" (hint-hint) that you have a Regular T which has both drains aligned with each other &amp; gravity's supposed to drop the water down the drain that's perfectly in the middle of both sinks. Like this.

 This might be what was previously there &amp; was fine. But, did you change anything like the vertical heights of anything? However, this is actually the wrong setup &amp; allows for a lot of debris build-up &amp; sometimes very poor draining.

-

The correct setup is to actually connect to the P-trap off to one side &amp; straight down from one of the sinks. Then you turn your T side ways (but preferably a WYE fitting...see last picture) to accept a long pipe from the other sink.


-

Here's the best setup, using a WYE. See how the fitting starts the gravity drop.


-

Anyway, I do realize this is likely an Off-The-Shelf Kit &amp; you won't be doing any of the above permanent type of setups. But, stoppers shouldn't be the problem if water's normally exiting the sink. I would take another look around the store &amp; ask about other or better double sink kits. Or, even see if they can help you screw one together of individual parts.

I think you just need to either match exactly what you had before or get a better kit &amp; even a different kit if another setup can work under your sinks. I have run into your problem a number of times before &amp; it was completely remedied by the 2 offset setups above.