#!/usr/bin/perl

use warnings;
use strict;

my $data;
{
        local $/;
        $data = <DATA>;
}
my @paras = split /(?<!\w)\n+/, $data;
print join "\n\n", grep { m/"/ } @paras;

__DATA__
Paragraph
to ignore.

Paragraph with "double quoted
text" to get.

Another paragraph
to ignore.

"Fully quoted
paragraph."