There are two solutions that come to mind for you.

1. You could use a persistent flag to create a state machine. All this means is you set a variable to true every time you see a line with "SQL". When you read in a line and the flag is true, you flip it and output that line. Something like:

   use strict;
   use warnings;
   
   my $flag = 0;
   while (my $line = <DATA>)
   {
       if ($flag) {
           print $line;
           $flag = 0;
       }
       if($line =~ /SQL/)
       {
           chomp(my $date = `date`);
           chomp (my $errcode = $line);
           print "$date - $errcode\n";
           $flag = 1;
       }#end of if
   }#end of while
   
   
   
   __DATA__
   Sun Sep 19 10:34:29 2010
   Online Backup for GMT failed.
   SQL2033N An error occurred while accessing TSM during the processing o
   +f a
   database utility. Reason code: 1.
   [download]

2. You could change your default record separator ($/), and use multi-line regular expressions (Modifiers) with capturing (Extracting matches) to extract the text you want. This is usually closer to how I end up doing this sort of thing, but is dramatically more complex.

As a side note, please wrap input and expected output in code tags as well as your code. It makes downloading it easier, and more importantly, it helps us make sure our regular expressions actually match what you mean, not what's written. See Writeup Formatting Tips.