Hi,
It may not be on your computer, it may be the server end.
That's not to say you should try to fix it.
Use or Create an Mac User account on your computer and set up Messages with the same Account details.
Does this work ?
If it does it will be some sort of file that the app accesses in your personal settings folders.
IF it does not work try a PRAM/NVRAM reset.
Hold down the ALT+CMD+P+R keys when doing a Restart of the computer.
Hold them down until you have heard three Start up Bongs then let the start Up continue.
Did this help ?
If not try a Safe Boot
This does have some limited functions but you should be able to Log in to your Mac user Account and use Messages.
Does this work ?
If not use the Finder's GO Menu whilst holding down the ALT key and select the Library that appears.
Navigate to Preferences
Find com.apple.ichat.plist and com.apple.imagent.plist and drag them to the Trash
Restart the app.
Does this help ?
After this you would need to look at the Keychain but I will wait for answers on the above
9:10 pm Wednesday; May 27, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad