Both sink and washer plumbed to the same drain which penetrates the basement floor slab. Drain is 1-1/2-2". Kitchen sink and toilet drain to a separate 3" copper pipe that also penetrates floor slab approx 6-7' away from washer sink drain. Never had a problem with sink filling and overflowing when the washer drains until recently. There what appears to be a clean out at about the midpoint of the 3" kitchen sink/toilet drain main and the basement washer/sink drain. Could it be the washer sink drain below floor slab is plugged. If so how can I fix it.
Would greatly appreciate professional advice. Home Depot and Lowe's don't seem to have anyone employed with the level of experience I am seeking.

Never seek advice from a sales person at a box store. All of your drains eventually come together into a main drain. Don't run any water for awhile to let it all slowly drain out, then carefully start backing out the plug in the clean out. Don't completely remove the plug but get it to the last thread so you can lift it slightly in case then is drain water standing in the pipe under pressure. Once the plug is out you can rod down stream toward the wall where the pipe exits. You should use asnake that is more than the entire length of the main drain. Is this city sewer or septic?