When I try to wash dishes, my sink basin slowly fills with water, and then water comes up the drain into the other basin, too. Why? How do I fix it?
Your drain is clogged up.
the clog is in the main line
I just unclogged the two bathroom sinks here a couple of weeks ago with a plunger. I filled one of the sinks a bit with water and kept working the hair/clog back and forth (as in the front and back side of the clogged gunk lump). I would do a few plunges then pull trash out of left sink then the right sink etc. as they each filled up with disgusting water until I thought I was close. Then I blocked one drain with the palm of my hand and plunged the other drain until i got the magic gurgle and they've been excellent ever since.
Sounds like the trap in the drain is clogged, or there is a clog below the trap. Most double basin sinks share a common p trap and if it or any piping below it clogs water will back up into both basins. You can try a plunger or a snake but the best tool short of taking the trap apart to clean it is called a zip it you can find them at most hardware stores or home improvement centers.
