Cannot convert from unicode to non-unicode error for Oracle 10g data source when switching to Run64BitRuntime = True

For packages that I have created to read Oracle 10g tables, that work fine with debugging in 32-bit mode, I get an error message on all string fields when I try to run in 64-bit mode.  An example error message is:

[OLE DB Source [1]] Error: Column "ACCT_UNIT" cannot convert between unicode and non-unicode string data types.

Another interesting warning included is:

[OLE DB Source [1]] Warning: The external columns for component "OLE DB Source" (1) are out of synchronization with the data source columns. The external column "ACCT_UNIT" needs to be updated.

I cannot even try to convert this data with a Data Conversion item because the (red) error is on the OLE DB Source item and stops there.  It doesn't matter what the destination is or even if there is a destination in the package yet.

I'm using Oracle Provider for OLE DB, Oracle Client version 10.203 for 32-bit and Oracle Client 10.204 for 64-bit.
Oracle is 10g on a UNIX 64-bit server and the data is not unicode.
I'm using SQL Server Enterprise 2008 (10.0.1600) on Windows Server 2008 Standard SP1 on a 64-bit server.

The packages work fine in 32-bit mode and the data is not unicode data.  When I change Run64BitRuntime to True in the Debugging Property Page, I get the error on the OLE DB Source item.  I also get the error when I schedule a package to run using the SQL Server Agent.