As far as I understand, as soon as I execute
Class.forName(\"net.sourceforge.jtds.jdbc.Driver\");
I initialize the application to use JTDS
You misunderstand. When you load a driver class with Class.forName(), that driver registers itself with the driver manager. You can do this with as many drivers as you want.
The first parameter of getConnection() is a URL that will uniquely identify the driver to use for that connection.
However, rather than getting connections directly from the driver manager, I recommend that you use a connection pool (such as Apache DBCP). This will let you get connections on an as-needed basis, and will provide some additional functionality such as warning you if you forget to return the connection to the pool.
You need to use DataSource. Configure a DataSource for each type of connection and the use the appropriate DataSource each time (e.g. via the proper DAO)