Here is a prior set of responses to basically the same question -
If you have any lower elevation drains like a shower/tub or floor drain and they are backing up too, then it is likely you have a partial blockage in your main sewer pipe downflow of the most downstream one that is backing up - commonly the lowest level drain(s) will start gurgling when the washing machine discharges, then as the sewer pipe gets blocked more also when toilet is flushed or bathtub is drained, then during showering, and finally at some point starts backing up and overflowing sewage from the lowest elevation drain during washer and maybe tub/toilet use.
In your case, since due to new washer, this is not an uncommon problem as many newer models pump out at a higher rate than they used to. Solution may be upsizing the drain from the washer to the main sewer pipe, or adding a vent pipe if it does not have one. I would say if you have lower elevation drains that are not gurgling or overflowing, then upsizing the drain pipe may be your only solution. If you have gurgling or overflow at lower level drains in addition to the adjacent sink, then I would have a sewer and drain contractor full-diameter rout out your sewer pipes all the way to the street (with a scraper blade) to clear out grease and soap scum accumulations, then run a camera - might as well have it run full length as long as pahying for any run at all, so total cost will be $250-400 range depending on pipe lengths to street. That run will tell you if you have any significant deterioration or root issues in your sewer pipe, which is always a good ting to know, plus will show you the twists and turns and vent and other connections in the system.
Unfortunately, there is another cause which I think I may be what your issue is - especially if the washer drain never overflows and any lower-level drains like tub/shower or floor drains are NOT overflowing. Washing machines are supposed to be connected directly to the main (3 or 4 ") sewer pipe, and I will bet someone cheated and ran the bathroom sink (and maybe toilet, shower, etc) and washer drains together before they come to the sewer pipe. During discharge the water level generally builds up a few feet in elevation in the washer drain - which increases the flow capacity quite a bit because of the added pressure. That is on reason why it sticks up above the floor so high. Unfortunately, when it does that backing up of water in the riser pipe it is backing up in the sink too, so if they are connected together instead of leading individually to the main sewer collector pipe, they should be isolated and run independently to it. This is likely to be a $250-300 minimum job (if pipes are all exposed in open walls and above slab level) to more like $500-1000 range if requires opening up floor slab to do it - PLUS repair of any walls and slab that had to be cut into.
You could also try a check valve on the sink pipe, but they generally do not work all that well and commonly leak at least some, plus become a place for future blockages, so I am not a fan of them unless required to stop backflow from the street sewer during blockages or major rains.
I would say if the aboe is your case, get a good plumber or sewer and drain man to come out and look at your pipes with a camera (unless clear to you how they run) and assess the situation.