I was referring to your iMovie library, thinking that if it were stored on an external drive there may be some problem with the drive. However, you are not using an external drive. Did you try creating a new library and seeing if you can save changes to a project created there? Just open iMovie while holding down the Option key and select to create a new library. iMovie will open in that library.
If still getting the error notification, try doing these diagnostic steps to see if any of them cure the issue:
Deleting Preferences
Open iMovie while holding down the Option and Command keys and select to delete preferences in the box that appears. iMovie will open in a new library. Reopen your old library to get back to your projects. Now see if everything is working properly.
Opening iMovie in a new user account
Create a new user account from System Preferences/Users and Groups and try to open and use iMovie in the new account. You can use the Guest account on your Mac for this purpose.
Safe Mode
You could also try booting up in Safe Mode (see Finder menu for instructions) and, after everything loads, immediately shut down and reboot in normal mode. Then see if iMovie works properly.
Updating iMovie
To update/redownload iMovie 10, drag your iMovie 10 app from the Applications folder into the trash, but do not empty the trash. Sign in to the app store and open your purchases folder by clicking on your name at the bottom of the app store side bar. You will see your purchases displayed, including iMovie 10. Redownload/update iMovie from there. If not successful, drag your present iMovie app out of the trash and return it to the Applications folder.
Resetting the System Management Controller
Try resetting the System Management Controller. It is simply a matter of booting up while holding down certain keys.
https://support.apple.com/en-us/HT201295
Resetting the NVRAM or PRAM
Try resetting the NVRAM or PRAM, although this issue would be fairly rare:
https://support.apple.com/en-us/HT204063
-- Rich