I'm confused on a number of issues reading your post. The sink "floods" but yet works fine and doesn't overflow. What do you mean by flooding then? Is the washing machine draining into the sink? Is it draining into the same pipe? Same plumbing stack? Are both being used at the same time when it "floods" or only when one is being used and if so, which one? Perhaps a picture?
Fair questions, and important ones. Personally, I suspect OP has a slow drain and the gray water from the washing machine is probably backing up into the sink where it's draining fast enough to the sewer line to keep things from overflowing to the point of spilling out, but not fast enough to actually handle the output of the washing machine without backing up. I'm going to assume that scenario with the following. If that isn't the issue, then the following might not be very useful.
If what was posted above on the details are accurate, it sounds like the common drain line between the kitchen sink and the washing machine drain has some amount of flow restriction either from grease, sludge, hair, roots, lint, mystery garbage, or some combination of the lot. Best thing to do is run a small powered auger down through the sewer line to clear it all out, and then run scalding hot water down the drain right after for about five minutes to help flush the remainder away.
Go to your local Home Depot and rent one of these:
https://www.homedepot.com/p/rental/Electric-Eel-Mfg-Co-Inc-and-General-Wire-Spring-Drill-Unit-Drain-Cleaner-25-x-5-16-SK-R-5-16EIC25/310643055Warning, the job will be messy. Wear cheap leather gloves (HD usually gives you a free pair with these sorts of rentals), grubby clothes, and keep anything you want even remotely sanitary plenty of feet away while doing this.
Here's a video on how to use one, though I'd suggest asking the rental clerk how to as well.
As for where to run the auger down? First question is, which drain is farther from the main sewer cleanout for the entire house? The kitchen sink, or the laundry room? If the laundry room is farther, just slowly feed the auger down the washing machine drain. If the kitchen is farther, don't run this particular auger down the sink drain, it'll be just a bit too big. Hopefully, you'll have
a cleanout under the sink to run the thing down through... otherwise, you'll need to take the curved trap off from the sink (use a bucket underneath, it's got water it it) and go in through there with the auger. Run it through a couple three times just to make sure you got it all.
If roots come back with the auger tip, ask about root killing chemicals down at the Home Despot to help keep them from growing back. It'll probably be a treatment you'll have to do a couple times a year if it's a problem. It'll also be a good idea to dump a couple gallons of boiling water chased with scalding hot tap water for about 5-10 minutes every few months from the kitchen end, too, to help clear out any greasy buildup as a maintenance preventative as well. Also, garbage disposals tend to cause most blockages off the kitchen drain lines. Just something to keep in mind.
It can seem a bit intimidating at first, but the sewer end of plumbing is pretty straight forward. If it doesn't work, it's clogged somewhere. Clean out the clog, problem is fixed. Mechanical methods are the most reliable ones for clearing a blockage, and plumbers in a bottle tend to cause more problems than solve.
More detailed instructions can be found with a little googling, and there's plenty of Youtube videos out there, and I'm sure others will help with some points that I might have missed.
Good luck!
