Our institution does not use a calendar tax year.  Several parts of the questionnaire refer to the tax year ending in 2006.  This will cause us to use information from our Forms 990 and 990-T that have a 2005 in the upper right-hand corner.  Is this correct?

Yes.  If your returns for the tax year ending in 2006 used the IRS forms dated 2005, those are the forms you should reference.