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.
Tornado victims: Do not rush to rebuild. Take your time and look for and get a good contractor. Or consider selling the property and moving to a home that is ready to live in.