I'm a little rusty, but this looked fun and I took a stab at it. Here's a convaluted, winding, unoptimized memory hog that generates a high scoring solution, but not always the highest possible score given arbitrary lines of input...

#!/usr/bin/perl -w
use strict;

my @lines=<DATA>;
my %hash = ();
my %highscores = ();
my @final_words = ();
my $final_score = 0;
my $longword = '';

for my $letter ( "a" .. "z" ) {
my $line = '';
 for (@lines) {
  $line = $_;
  my @words = split;
  for my $word (@words) {
   my $tmp = scalar grep {/$letter/} split (//,$word);
   if (!$hash{$line}{$letter}{highscore} || $hash{$line}{$letter}{high
+score} < $tmp) {
    $hash{$letter}{$line}{highscore} = $tmp;
    $hash{$letter}{$line}{word} = $word;
    $hash{$line}{$letter}{highscore} = $tmp;
    $hash{$line}{$letter}{word} = $word;
   }
  }
 }
 $highscores{$letter} = 0;
 $highscores{$letter} += $hash{$letter}{$_}{highscore} for keys %{$has
+h{$letter}};
 if (!$highscores{bigletter}{score} || $highscores{bigletter}{score} <
+ $highscores{$letter}) {
   $highscores{bigletter}{score} = $highscores{$letter};
   $highscores{bigletter}{letter} = $letter;
 }
}

my $bl = $highscores{bigletter}{letter};
print "Letter with highest score is \"$bl\" with $highscores{bigletter
+}{score} occurrences\n";

for my $line (keys %{$hash{$bl}}) {
 push @final_words,$hash{$line}{$bl}{word};
}
print "\nWords chosen:\n";
print "$_\n" for @final_words;
$longword .= $_ for @final_words;
my @final_letters = split (//,$longword);
for my $letter ( "a" .. "z" ) {
  my $count = grep {/$letter/} @final_letters;
  $final_score += ($count * $count + $count) / 2;
}
print "\nFinal score: $final_score\n";

__DATA__
alabama arkansas alaska delaware hawaii indiana kansas montana delawar
+e
sazerac sangrita
radiator gastank engine heater fender wheel detent battery clutch mirr
+or window
alloy amalgam vanadium copper steel
rutabaga limabean cress carrot sorrel squash cabbage pepper lettuce be
+et leek celery endive rhubarb parsnip pumpkin
azalia camellia dahlia gardenia gentian vervain canna hepatica bluebel
+l anemone oleander
lasagna macaroni pastina gnocchi tortelli alfabeto
mascagni britten menotti
unamas tacotime pizzahut tacobell panago tacomayo edojapan hardees
caramel marzipan taiglach taffy brittle fondant toffee dragee

__OUTPUT__
Letter with highest score is "a" with 26 occurrences

Words chosen:
alabama
unamas
sazerac
radiator
lasagna
amalgam
caramel
azalia
mascagni
rutabaga

Final score: 457
[download]

Note the score is less than the shown maximum of 474 for this data set.