Docker is running now.
Many different troubleshooting steps were taken but the last steps that I took before it was successful was the following:
1.Uninstall Docker Desktop
2.Restart computer
3.Install Docker Desktop (make sure to install it with admin rights)
I had uninstalled/reinstalled docker many times and separately restarted my laptop many times but am not sure that I followed the exact steps above in that order.
After this, I was also able to enable integration with my additional WSL 2 distros.
Hope this helps someone else who is having a similar problem.