If ExQuilla is claiming that your password is incorrect, there are a few things that you can do to test it.
Authentication in ExQuilla relies on four pieces of information being correct: 1) your EWS URL, 2) your user name, 3) your domain, and 4) your password. The first three items may be viewed in the account settings (Tools/Account Settings/(your Exchange account)/Microsoft Exchange Server). If the username resembles an email address, then usually the domain is blank.
If your account used to work, there may still be entries in your ExQuilla log file that shows the EWS URL, username, and domain that was used for a successful login. Here is an example from my log file of a successful connection with EWS URL "https://email.hostaccount.com/EWS/Exchange.asmx", username "email@example.com", and blank domain:
2013-11-11 10:30:42 extensions.exquilla.webservices DEBUG asyncCall uri https://email.hostaccount.com/EWS/Exchange.asmx user firstname.lastname@example.org domain null
2013-11-11 10:30:42 extensions.exquilla.webservices DEBUG httpSoapTransportCompletion handleEvent type load username rkent%40mesquilla%2Ecom
2013-11-11 10:30:42 extensions.exquilla.webservices DEBUG transferComplete to URL: https://email.hostaccount.com/EWS/Exchange.asmx
2013-11-11 10:30:42 extensions.exquilla.webservices DEBUG contentType: application/xml status: 200 statusText: OK authCount: 0
You may view your existing username and password in the password manager (Tools/Options/Security/Passwords/Saved Passwords) under the entry for your EWS URL. Try clearing the password there and restarting Thunderbird, which will force a new request to enter a password.
One test that you can do of these pieces of information, independent of ExQuilla, is enter the EWS URL into a browser. It should ask you for a name and password, and if successful show you a page of XML-language information that starts with:
If you can get that, it means that you have successfully connected to the Exchange Server. In almost all cases, the same information should also allow ExQuilla to connect successfully.
You can also confirm that the username and password works with Exchange Web Services using Microsoft's connectivity test site (https://testconnectivity.microsoft.com/). Choose "Microsoft Exchange Web Services Connectivity Tests/Synchronization, Notification, Availability, and Automatic Replies" and make sure that you can connect using the username and password that you expect.
Unfortunately if authentication fails, there is little that we can do to help you. The ExQuilla logs will have entries like "extensions.exquilla.base INFO html error <Status: 401 Message: Unauthorized>" and little else of value. For us to help you overcome this, we would need to have valid credentials to login to your Exchange server (which could be a test account). This is rarely an option, which makes it difficult for us to help beyond asking you to try different things.