Is there a way that my iMessages can be going to a device that is not signed in to my Apple ID?
There are multiple places in which your AppleID can be used on a given device: iMessages, FaceTime, iCloud, App Store, etc. It sounds like what is going on is that the iPad is still logged into your AppleID for iMessages, but perhaps not for other items.

Perhaps the easiest way to handle this is to change your AppleID password. You can do this on Apple's site: https://appleid.apple.com. After that is done, you will then need to go through each of your devices and update the password that is used for the App Store and Messages/FaceTime.