Hi there uniquelise,
Thanks for using the Apple Support Communities. I understand you’re having some issues using FaceTime on your Mac. I love using FaceTime to stay in touch with my friends and family, and I’ll be happy to help with this.
Based on the fact that you can’t use Photo Booth because FaceTime is using the camera indicates you have FaceTime installed correctly and running. To verify do you see the FaceTime icon in the Dock? If not, click on the Apple logo in the top left corner of the screen and click Force Quit. Do you see FaceTime listed there? If so, quit the app then open it back up from the Launchpad (the grey circle with a rocket in your Dock).
Once FaceTime is open, you can use these steps to help you sign in with your Apple ID and make a FaceTime call:
Use FaceTime on your Mac
If you continue to have trouble using FaceTime after that, take a look at this article for ways to resolve the issues:
If FaceTime isn't working on your Mac
Best wishes.