A new licence will not affect the database connection.
If your systems cannot connect to the database, you've changed something else. Either their connection settings are now incorrect because you've edited the settings files (confluence.cfg.xml and dbconfig.xml in their home directories) or you've told the database server not to allow access from those servers, or you've put a network security thing in place that is blocking the route.
Check the networking (on the application server, not your machine) with a simple "telnet database-server 1433" (assuming your database is listening on 1433 - if you get a block of text saying you need to log in, then the network is ok. If not, then the error message should tell you what the problem is.
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