How do I save my edits on Lightroom in a way that allows me to access them from a different computer to continue editing? I want to use two different computers to edit the same image on lightroom.

You can do this with LR CC if your total images isn't so much as to fill up your hard drive with the sync data, or you can put your images and catalog on an external drive, and as long as you name the device the same on both computers or have the same drive letter on both computers it should work with the desktop LR Classic CC.  Just make sure you've shut down LR completely -- and the process has exited, after several 10s of seconds for me, before you eject the drive.