I am not sure what you are doing to cause this problem as no-one else has managed to have a problem with verifying their accounts (that I know). I do know that the error messages you are reporting are because the account name you have entered doesn't match the authorised twitter account username so maybe you haven't verified the twitter account correctly or you waited too long before entering the PIN code that Twitter gave you?
I don't know how you are getting a load of underscores _______ back? Or is that supposed to represent something else?
I would suggest deleting the accounts in the plugin admin area and then try re-adding them and then re-verifying.
If that doesn't work try uninstalling the plugin and re-installing.
If that doesn't work I would suggest seeing if any other plugins are causing the problem by disabling them one by one as sometimes plugins collide with each other.