I am still learning about syncing computers, but I can provide some assistance. If you have all 3 computers on the same Workgroup and can access them from each desktop using Explorer, then you can share the data using windows or a third party app such as FreeFileSync.
Resilio Sync does work continuously in thebackground. When you set up the folder to sync, be sure to check Read and Write. I think you need to set up a sync link from computer 1 to computer 2, and a link from computer 1 to computer 3. As long as you only work on one computer at a time, a) a change to computer 1 will sync to computer 2 and 3; b) a change to computer 2 or 3 will sync with computer 1; c) and a change to computer 2 (or 3) will sync with computer 1, which should then sync back to computer 3 (or 2).
Obviously I'd suggest testing this settup on a sample folder.