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map {
   s/.../.../seg;         <-- Escapes the index.
   $form_data{$_} =~ ...  <-- Should be using unescaped index.
   ...
}

$form_data = join '&',
             map { my $key = $_;
...
                   $val =~ s/([^A-Za-z0-9])/sprintf("%%%02X", ord($1))
+/seg;
                   "$key=$val" }
             sort keys %form_data;

use URI::Escape qw( uri_escape );

...
                   my $val = uri_escape($form_data{$_});
                   "$key=$val" }
             sort keys %form_data;

use URI;
use URI::QueryParam;
...

$uri->query_param($_ => $form_data{$_})
   foreach sort keys %form_data;