I get multipe values for my keys and values appeared more than one

Each hash key only holds one value. In this case, the value is a reference to an anonymous array. So your question really is "How can I remove duplicate elements from a list or array?"

This is a FAQ. You'ld process each anonymous array and remove duplicates, or create a new anonymous array holding unique items and assign that to the value slot of each hash entry.

If you don't want to reassign to the value slot - because, say, the reference is stored elsewhere too, and you don't want that link to be destroyed - you could use splice to edit the anonymous arrays in-place, like this:

$HASH1 = {
    'Alabama' => [
        'Andalusia', 'Anniston', 'Clanton', 'Eufaula',
       'Auburn', 'Bessemer', 'Eufaula', 'Auburn', 'Bessemer',
    ],
    'California' => ['Barstow','Barstow'],
    'Georgia' => ['Darien'],
    'New York' => [
        'Amsterdam','Coney Island','Coney Island',
        'Becon','Becon',
    ],
};
for my $key ( keys %$HASH1 ) {
    my $arrayref = $HASH1->{$key};

    my %seen; # empty at each iteration
    $seen{$_}++ for @$arrayref;

    # process array from the end towards beginning

    for my $index ( reverse 0 .. $#$arrayref ) {
        if ( $seen{$arrayref->[$index]} > 1 ) {
            $seen{$arrayref->[$index]}--; 
            $removed = splice @$arrayref, $index, 1;
            print "key $key: removed '$removed'\n";
        }
    }
}
dd $HASH1;
__END__
key Alabama: removed 'Bessemer'
key Alabama: removed 'Auburn'
key Alabama: removed 'Eufaula'
key New York: removed 'Becon'
key New York: removed 'Coney Island'
key California: removed 'Barstow'
{
  "Alabama"    => [
                    "Andalusia",
                    "Anniston",
                    "Clanton",
                    "Eufaula",
                    "Auburn",
                    "Bessemer",
                  ],
  "California" => ["Barstow"],
  "Georgia"    => ["Darien"],
  "New York"   => ["Amsterdam", "Coney Island", "Becon"],
}
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Note that in the OP, in the anonymous array for the key New York you have the entries Becon and Beacon which are different - a typo, I guess.