A few comments:

You never use $check. One thing I've taken on as a habit is to declare the variables I use as 'my' when I use them. When you pre-declare them in one string as you have with my ($ua, $req, $chrom, $start, $end, $db, $check);, you increase the chance of error when you remove/add them later, as you have in this script. One possible solution is as follows:

my $ua          = LWP::UserAgent->new;
my $chrom       = "7";
my $start       = "117287204";
my $end         = "117475903";
my $db          = "hg11";
my $url         = "http://genome.ucsc.edu/cgi-bin/hgText";
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(I also like to indent for readability, but that's more a matter of personal style.)

You also are not properly separating your POST object. It should look something like:

# Indented again
my $request     = POST $url,
        ['hgsid'        => "6765675",
         'g'            => "htcGetDna2",
         'c'            => 'chr'.$chrom,
         'l'            => $start,
         'r'            => $end,
         'db'           => $db,
         'hgc.dna.out1' => 'lc',
         'boolshad.hgc.dna.rc' => '1',
         'Submit'       => 'Submit'
        ];
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Note the use of the single quotes around the key, which was missing in your original script. Adding the above fixes allows your code to work as intended.