How to stop iMessages from coming to my computer

On your Mac, open Messages. Go to Preferences>Accounts. Select your Apple ID and then untick "Enable this account".

Add or remove your phone number in Messages or FaceTime - Apple Support

Stuart423 wrote:

Dang...I'm guessing I should take time to read the question, IdrisSeabright. This has not been a stellar week for me...

Not a big deal. And, the nice thing about a bad week is that it has to end. And you get a whole new one. 😉

Open system preferences (the grey gear wheel on the dock, or on the Apple menu top left) and select General. Uncheck the 'Allow handoff...' entry.

Stuart, I don't believe Handoff has anything to do with SMS/Messages.

Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch - Apple Support

Glad it helped!

Dang...I'm guessing I should take time to read the question, IdrisSeabright. This has not been a stellar week for me...

Yup.