I'm puzzled by the first "if" condition at the top - you do the same thing no matter what the "request_method" is, so why is the "if" statement there at all?
Apart from that, when you call the "header" sub, you call CGI->new, and then when you call the "display" sub right after that, you call CGI->new again -- but it's not clear to me that the second call will yield all the same stuff that the first one did, so that might have something to do with the problem.
I'd suggest that you restructure things a little bit at the top, like this:
# You do have taint mode turned on, don't you? and the next two lines,
my $cgi_obj = new CGI;
my $page = $cgi_obj->header( 'text/html' );
$page .= display( $cgi_obj );
my ( $query ) = @_;
my $html = qq~
... (initial boilerplate stuff for the page) ...
# run your dbi queries...
# append more html stuff to $html as you go, and finally:
$html .= qq~
Another problem is that when you format your paging links, you are printing "page", followed immediately by a number, (e.g. "page1", or "page2"), and that becomes the name of the url parameter. But when you handle the cgi query params on a subsequent (GET or POST) request, you're looking for a parameter named "reqpage", which was never put into your page links. That would certainly lead to disappointment.