You may still use an old ODP.NET for newer database version.
So if your database is being upgraded to 11g, there is no requirement that you must upgrade your version of ODP.NET.
Also as far as I know, the latest version of Oracle.DataAccess.dll that uses .NET famework 1.1 is 11.1.0.7.0, so you may try this one.
So first thing I sugget is to test the application against the new database without changing nothing except the connectionString.