For starters, don't worry about the old, encrypted password in the DBALIAS.INI file. When moving to a new SQL server, you'll get similar error messages, but will be prompted for SQL credentials when logging in to GoldMine. Simply enter the new SQL credentials and check the Save Settings option and that part should be fine. (Manually changing the password in the DBALIAS.INI file will not help in the least.)
THIS is very likely your key issue: SQL Server does not exist or access denied.
Are you trying this FROM the new SQL server itself, or from a DIFFERENT computer?
This error message can happen if trying from a different computer (than the SQL server) and if either the SQL server (most often) OR the computer you are on, have Windows Firewall settings inhibiting SQL traffic. If this is the case, you will need to configure the firewall in question to allow SQL traffic to flow (you can Google how to do this).
I've also seen this same symptom, when trying from a different computer, where the Windows Firewall was NOT the issue, but the issue was resolved by simply STARTING the "SQL Server Browser" Windows Service on the new SQL Server. This is not ALWAYS the solution, but I have seen it prove to be such on more than one occasion.
Anyway, those are two possibilities.
Hope this feedback helps. Please let me know if I need to clarify anything.