Frank B,, you are a writer. You are the primary/only user of both computers, but you use only one at a time. Am I correct so far?
Your manuscript is important, but you don't necessarily have to retain every change you make during the writing process; what you really need is to be able to use the most up-to-date version on whichever computer you happen to be using. But you may also want to keep one or more preliminary versions where you can access them, in case you need to go back at some point. Correct?
All you need is a portable storage device -- a USB hard drive, SSD or even a flashdrive. -- with the version(s) you need, and which can be plugged into either computer as needed. Save each older version with XXXXXxx, where XXXXX is the main filename and xx is a two-digit revision number: XXXXX01 is the first "old" version, XXXXX02 the second, etc. The filename XXXXX (without a revision number) is your "working" manuscript file, which will be copied to storage using "Save As" and appending the next available revision number when you archive it prior to continuing or editing your work; XXXXX (the working copy) is the only version you need to keep on whichever computer you are using, and it can be copied to the internal drive on the other computer as needed. The revision numbers give each version it own unique filename as far as the computers are concerned, so they don't get over-written; just change the revision number before you "Save As" to storage at the end of the workday.
Personally, I'd keep XXXXX (the current working copy) on a portable storage medium as well, but not necessarily the same one as your earlier versions. You might even make two copies, just in case you misplace one.
Because word processor documents are relatively small -- generally megabytes rather than gigabytes -- a decent-sized flash drive can hold multiple earlier versions of even a long work-in-progress along with loads of reference notes, still images and even videos you might need while writing. A much smaller flashdrive can hold the final work, which can get go into a safe-deposit box at the bank or an office safe in your home workspace.
By keeping all the needed manuscript versions and reference materials on a flashdrive or other medium, you can easily copy everything to the internal drives on either computer (or both, just one at a time), *and* provide a backup you can slip into a pocket wherever you happen to be.
If you're cautious-bordering-on-paranoid, make extra copies while on the road, keeping one on your person; use a NAS (network-attached storage) device at home; and arrange to stash a copy somewhere offsite (a trusted friend's home, perhaps) now and then.in case of fire or natural disaster.
And don't write off cloud storage completely. Simply use a service such as iDrive or Carbonite -- continuously at home, and whenever you can on the road. Many coffee shops and restaurants (notably) Starbucks and McDonald's) offer free public Wi-Fi; these services aren't especially secure, but you can always encrypt your files before uploading, and the employees are used to seeing people using their laptops while they sip or dine.