Hi,
Start with what Eric says. These two .plists hold account info about the app.
If that does not help check the Apple Icon Menu > About this Mac
Click twice on the line that tells you the current OS
It will change to the build number and then show the Serial Number.
If the Serial Number is present go to Applications/Utilities and open the Keychain Access item
Have Login set top left and Passwords lower left.
In the right hand pane look for the items that start IDS.
one will be IDS: yourAppleIDemail-AuthToken where yourAppleIDemail is the Apple ID used in iMessages.
Delete this items and restart Messages.
This should cause Messages to refresh it's details about the Login to the server.
If it is recreated it may be enough to kick start the account again.
More likely is it will return to the current state (partial login) where you cannot send iMessages but appear to be logged in.
If this happens it tends to prove the issue is at the Server end.
If this appears to be the case you have to contact Apple Support and explain you have an Accounts issue (Free to Fix).
Most likely you will have to ask for a level 2 person as unfortunately the Level one people can be a bot too much led by the scripts they work from (Aimed and Hardware and Software issues and Apple Care and payment).
8:36 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