The following returns all the sets of 5 uniqe letters that agree with the rules (two and only two letters are common between the set of letters and each of the words). Note that I've also added a rule, that every set of letters must contain ONLY letters included in the input words.

use strict;
use warnings;

my (%l, @a, $bad, $w);
my @w = ('bumps','seams','domes','shake','pokes','dukes');

for (@w) { $l{$_} = () for (split //, $_); }
permute(5, '', \@a, sort keys %l);

for $w (@a) {
    $bad = 0;
    for (@w) {
        if (both($_, $w) != 2) { $bad = 1; last; }
    }
    if (!$bad) { print "$w\n"; }
}

sub permute {
    my ($d, $w, $a, @l) = @_;
    if (!--$d) { push @$a, $w.$_ for @l; }
    else { permute($d, $w.$l[$_], $a, @l[($_+1)..$#l]) for (0..($#l-$d
+)); }
}

sub both {
    my ($w1, $w2) = @_; my %l; my $c = 0;
    $l{$_} = () for split //, $w1;
    for (split //, $w2) { $l{$_} = 1 if exists $l{$_}; }
    for (values %l) { $c++ if $_; }
    return $c;
}
[download]

Sloppy code in places probably, but it does the job. Returned values are:

abdep
abeou
adkmp
akmou
[download]

Problem is, I can't see a way to arrange any of these into an english word, unless I'm missing a possibility on the first set. Obviously, some of the letters have to be used twice, or letters not in any of the original words have to be allowed. Back to the drawing board :\

EDIT: Ohhh, you changed the puzzle. Time to see if I can get this one to work...

Nope, doesn't work with the new puzzle either. I'd delete my post at this point, but I should leave it here in case someone can gain insight from my mistakes.

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