I think the OP wanted to output the count of widgets at the top of the database section which was counted. That is, something more like this:

my $widgetsFound=0;
my $databaseName;
my $databaseSection;
while( <> )
{
   if( /^DATABASE:/ )
   {
      if( $databaseName )
      {
         print $databaseName;
         print "Widgets found: $widgetsFound\n";
         print $databaseSection;
      }
      
      $databaseName = $_;
      $databaseSection = "";
      $widgetsFound = 0;
   }
   else
   {
      $databaseSection .= $_;
   }
   
   if( $something )
   {
      $widgetsFound++;
   }
}

# print the last database section that was parsed
if( $databaseName )
{
   print $databaseName;
   print "Widgets found: $widgetsFound\n";
   print $databaseSection;
}
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Updated: to correct errors noticed by Animator (who answered the original question first as well :) ).