I have a double sided kitchen sink and both side are filling up with water. I do have a garbage disposal we use often and a dishwasher. They both have separate lines. I took apart the pipes to drain them and find out why both side of the sink were filling up. I found some small plastic from a medicine cap. I removed the plastic from the pipes and then the plastic I found in the inside of the garbage disposal. Still both side filling p with water. Should I snake the back pipe running into the wall or just call a plumber? Please help. Thanks in advance.

Should I snake the back pipe running into the wall? That's what I'd do.
Remove the "J" bend from the trap and send a snake into the wall. The good news is that you'll only have to nput out 6 feet. Good luck, Tom