First off, confirm that the network drive on the computer that crashed is accessible, with the same drive letter as before. For example, if you previously mapped the network drive under H:, make sure it hasn't changed to something else (e.g. Z:).
Uninstalling Clippings (or any Firefox add-on) is essentially a "full reset," as that removes all its settings and data from Firefox.  There is a possibility that your Firefox profile may have gotten corrupt when your computer crashed, in which case you may want to start over with a new profile.  You would need to reinstall Clippings (and other add-ons), and turn on Sync Clippings from the add-on preferences page.
That said, there is one more thing you could try:
If you haven't already, make a duplicate copy of the clippings-sync.json file, just to be safe.
On the computer that recently crashed, launch Firefox, then navigate to about:debugging. Look for Clippings in the list of installed add-ons, and make note of its Internal UUID.
Then open your Firefox profile folder (in Firefox, Help -> Troubleshooting Information, then under Application Basics, click the button next to Profile Folder). Go to the "storage/default" folder and look for a folder named "moz-extension+++f<Clippings internal UUID>".  You will need this in the steps below.
Go back to Firefox, then uninstall Clippings, and then exit Firefox.  Uninstall the Sync Clippings Helper. To be sure that the uninstall is complete, check that the Sync Clippings settings file is gone (look for the file named "syncClippings.ini" under C:\Users\<your user name>\AppData\Sync Clippings).  Go back to the Windows Explorer window where the Clippings storage folder was selected -- that folder should now be gone.  
Restart your computer. At this point, you should now be starting over with a clean slate. Once your computer is booted up, reinstall Clippings and the Sync Clippings helper app, and turn on Sync Clippings from Clippings preferences.
Hope this helps,