SQL Server connection lost when SA password is changed

Without seeing any of the source code, this is hard to troubleshoot. However, it sounds like the password to the administrator account (and here I'm assuming you mean the "SA" account) is hard-coded into whatever client-side application you're using. When you change the password on the server, the clients all fail because they are unable to login. Check the SQL error log to confirm this (look for authentication failures). You may need to re-write the client-side application to make the password configurable instead of hard-coded.