What can we do about a separate drain from our washing machine to our septic system that has failed?
Don't auger it. Trust me, that won't fix the problem.

I had exactly the same thing happen at my own house a couple of weeks ago, only mine made it for nearly twenty years before it developed problems. In my case, I knew that the drywell was "somewhere" in the back yard, nothing else except that it had to be lower than the exit pipe through the wall.

So... I dug. I found one partially-collapsed pipe (inappropriate installation), one gradual bend (flexible pipe) that had completely clogged solid with detergent (from the previous owners, since we only use liquid detergent), and a collapsed lid on the drywell itself... besides only one tiny (24" long) leach field from the drywell.

None of this could have been in any way helped with an auger, although (since I knew I'd be digging up the rest of it anyway) I did auger a little in a couple of places, just to temporarily relieve backpressure on the drain plumbing inside the house. Augering "downward" from that detergent-clogged bend only pushed the detergent clog down toward the drywell - I later dug it out of the drywell along with some twenty gallons of crushed rock and the remains of the lid. Augering "upward" from that bend only knocked loose the detergent clog, dumping it into my hands.

Augering ONLY pushes the problem FURTHER down the line, it DOES NOT remove it. The only thing augering CAN remove is hair, because that hair wraps around the auger.

Had I not dug up the drywell, discovered the collapsed lid, and had to empty it out... that detergent clog I augered "downwards" would have eventually worked its way into that tiny leach field, where it would have been an even worse problem.

I found no need whatever to put in a whole new drain. I DID patch the partially-collapsed spot, and I DID replace the drywell lid, and right now I'm in the process of laying in an additional 10' of leach field in the hopes that, since 24" served for 20 years, 144" will hopefully last for the rest of our lives. Note that I DID NOT dig up the entire system, only the spots that needed digging to expose problems.

My first exploratory hole was 23' out from the house because I knew I could feed 23' of measuring tape into the pipe before it stopped. I also knew the angle from perpendicular-to-the-wall because I could SEE it in the remaining 2' of measuring tape... so that first exploratory hole (where I found collapsed pipe) ended up being only 24" square, hand-dug with a shovel. The second exploratory pit, which is where I found the clogged bend, was another 12' further out - also hand-dug with a shovel.

I did rent a backhoe ($250US/24 hours) to dig the trench for my additional leach field. I may be crazy, but I'm not NUTS.

If your drywell - or the plumbing leading to it - is clogged, you need to determine where the clog is, dig up THAT SPOT, determine the cause of the clog, and devise a means of correcting the problem. You may find something collapsed, you may find a wad of "something" that SHOULD NOT be pushed further, you may possibly even find that your drywell or its leach field is insufficient given your soil type and depth.

Oh, yes. I should also mention that the first half of this drywell's life was prior to our ownership, and that the previous owner had a nasty "puppy farm" here, and that PART of the contents of the drywell and leach field was very old dog manure and doghair (the drywell also served a floor drain in one indoor kennel). I poured half a box of Rid-X (locally available brand, not an endorsement, other brands work equally well) into the drywell to digest the remaining manure &amp; doghair... and that DID help a lot, changing the greywater from solid black to translucent blue-grey.