Others may have different advice, however if you Really, absolutely, etc, are stuck with the usb stick for now my suggestion would be to work as follows. Note, This does require keeping CLOSE Documentation of your actions, perhaps on a notes application you keep open on each computer.
Have your save to location for your project on your local hard drive.
When you are ready to close your project, do manual Save, followed by a Save As to your thumb drive with a unique name say [project_name]2304 (todays date). Ensure all drive activity is complete before using the eject drive function.
Also ensure you have set backup on close Project, with it set to zip backup (it’s harder to accidentally open a zip project).
Then insert the thumb drive in the second computer when you are ready to work on it.
Immediately on opening the project, do a save as [project_name] to that computers normal project save to location. It will overwrite the last saved project.
Continue to work on the project.
When you are ready to close, do a save, then a save as to the thumb drive [project_name]date (and an a,b etc if you are doing multiple saves on the same date)
Working this way means you are not copying projects and potentially screwing up, BUT, you must scrupulously document the file names to make certain you don’t mix them up and overwrite later work. If something goes wrong, whatever you do, do not start opening projects looking for the latest. Sit down with your notes and carefully decide which is your latest save, and which us your latest zip backup. Protect Them all costs. With what you think is the latest save Copy and rename the entire .scriv folder (don’t open it and copy any internal files/folders - just copy the entire .scriv) so you have a saved version of it before you then open it and check. There’s a whole similar process To opening the zip backups.
There is a MUCH simpler solution if both computers gave internet access. Get a Dropbox account, set it up on both computers and simply point to that local Dropbox folder as your save location. Then all you gave to do us make certain the sync has finished before closing down each computer and that you only ever have the project open on one computer at a time.
I know which solution I’d use… Dropbox wins every time. Note you CANNOT do that with any other cloud service.