When so many things can go wrong all you do is try, try, try

Seriously, how can you handle all those exceptions without going nuts? Have I read one too many articles on exception handling or what? I tried refactoring this a couple of times and each time I seem to end up with something even worse. Maybe I should admit exceptions do happen and simply enjoy coding just the happy path? ;) So what's wrong with this code (besides the fact that I was lazy enough just to throw Exception instead of something more specific)? And by all means, don't go easy on me.

public void Export(Database dstDb)
{
    try
    {
        using (DbConnection connection = dstDb.CreateConnection())
        {
            connection.Open();
            DbTransaction transaction = connection.BeginTransaction();
            try
            {
                // Export all data here (insert into dstDb)
                transaction.Commit();
            }
            catch (SqlException sqlex)
            {
                ExceptionHelper.LogException(sqlex);
                try
                {
                    transaction.Rollback();
                }
                catch (Exception rollbackEx)
                {
                    logger.Error("An exception of type " + rollbackEx.GetType() +
                                      " was encountered while attempting to roll back the transaction.");
                }
                throw new Exception("Error exporting message " + Type + " #" + Id + ": [" + sqlex.GetType() + "] " + sqlex.Message, sqlex);
            }
            catch (Exception ex)
            {
                try
                {
                    transaction.Rollback();
                }
                catch (Exception rollbackEx)
                {
                    logger.Error("An exception of type " + rollbackEx.GetType() +
                                      " was encountered while attempting to roll back the transaction.");
                }
                throw new Exceptio