They make a perforated plastic dry well (dry bed) liner to put in the hole you dug. An alternative is bricks laid not without mortar with a prefabricated concrete or steel lid. Then just about anything (PVC pipe suggested) can be used to re-route the washing machine drain line to that.
Some city codes may prohibit the use of a dry well for anything other than plain water such as rain water.
You really should not need to re-route your washing machine drainage from your septic tank unless you are experiencing septic system problems.
Do you get coverage you can count on? From the bag of fertilizer? From your lawn sprinklers?