Kitchen sink backfeeds

Sorry for the dumb question, but I have very very limited plumbing experience, and it's all getting the water to the house, not getting it out.
My house was built in 1975 and the sink is a side by side with garbage disposal. Lately it has been backing up, and when I hit the garbage disposal it fills up the opposite side. It is also very slow to drain. I took a quick peek under the sink and the drains don't look immediately connected. I tried drain-o down both drains with little to no improvement.

Where should I start?

drains

With a good plumber who has the necessary drain cleaning equipment and the experience to use it properly.

Draino doesn't work that often, plus, it can eat up a garbage disposal. It also can mess up a snake or injure a plumber (or you) if the pipes need to come apart for cleaning. You have a clog, and it will probably take a snake and a plumber to fix it. Stop putting chemicals down the drain.

Draino disolves grease and hair. Period. It won't disolve the little bits of food stuck in the pipe from your disposal or correct backpitch problems or anything like that. HJ is right, call a local plumber with drain cleaning experience and good equipment. You'll probably find that a local pro costs less than the national "Rooter" chains