Does signing out of FaceTime disconnect your Apple ID from a specific device?
Yes - tapping the sign out button disconnects FaceTime. The messages app uses a different account unless you set them all up using iCloud preference or a setup assistant screen to log into all icloud services with one Apple ID. 

As long as you signed out of the FaceTime portion of the settings app - more deliveries won't happen to that device. 

as long as you didn't sign in to the iCloud settings or anywhere else - you should be safe with no data of yours on that device other than for the moments you allowed it.