It sounds like you should be using package configurations.

First, I would set up a standard set of package-level variables with the same naming convention in each package. These will be your "Source Server" and "Source Database."

Next, use the Expressions in the Connection Manager's property sheet to assign the variables to the Connection Manager's server and database properties. (For OS file connection managers [flat, Excel, etc.], set the "connection string" property directly from the variable.)

Finally, set up package configuration(s) to set the package variables. You can use a file-based configuration file-based for this, and then you deploy the same package with server-specific configuration files.

I use the variables because (a) easier to set variable values, (b) one variable may set multiple connection managers, and (c) its easier to use consistent names and reuse configuration files for multiple packages.

In BIDS, the package configuration is part of the SSIS Project and gets used automatically. When you deploy to the server, you need to specify when you run a package that it needs the configuration files applied and where to find them; otherwise it will work with whatever default settings are in the package. A SQL Agent SSIS job step makes it relatively easy to specify the configuration file.

You can also use multiple configurations with a package, and multiple types. So, for example, you can have one permanent environment variable on the server that specifies the "data source" for a connection manager that is used for "SQL server table" package configurations. [You might also be able use "(localhost)" for servername here instead of an environment variable.] All the package-specific and volatile settings are in that SQL Server table. If you need e-mail (SMTP configuration managers), these could be single share configuration file where the server admins can maintain it. If you pick the right combination, you should be able to keep everyone reasonably happy.