These are the steps that I used to run Docker without using Docker Desktop.

Ensure you are using WSL 2 from PowerShell.

wsl --list --verbose

If you are, then open up a bash shell in WSL 2.

Completely uninstall Docker.

dpkg -l | grep -i docker

sudo rm -rf /var/lib/docker \
  /etc/docker \
  ~/.docker \
  /var/run/docker.sock \
  /usr/local/bin/docker-compose
  
sudo rm /etc/apparmor.d/docker

sudo groupdel docker

Then follow the steps for installing docker using the repository here https://docs.docker.com/engine/install/ubuntu/ Those steps will look something like this:

sudo apt-get update

sudo apt-get install \
  apt-transport-https \
  ca-certificates \
  curl \
  gnupg-agent \
  software-properties-common

curl -fsSL https://download.docker.com/linux/ubuntu/gpg | sudo apt-key add -

sudo add-apt-repository \
  "deb [arch=amd64] https://download.docker.com/linux/ubuntu \
  $(lsb_release -cs) \
  stable"
  
sudo apt-get update

sudo cgroupfs-mount

sudo service docker start

sudo docker version

Resources

https://stackoverflow.com/a/56051167/1108891

https://docs.docker.com/engine/install/ubuntu/

https://askubuntu.com/a/1021506/500631