my $sql = <<ENDSQL;
select ...
  from ...
       ...
  where so.value = ?
ENDSQL

for my $lookup ( sort keys %dbsrv ) {
    # ... make the connection as you do now
    do { warn "server $lookup: connect failed\n"; next }
        unless $dbh;

    my $sth = $dbh->prepare( $sql );  # "prepare" once
    do { warn "server $lookup: prepare failed\n"; next }
        unless $sth;

    for my $key (sort keys %list) {
        for my $rpt ( sort keys %{$list{$key}} ) {
            my @drec = @{$list{$key}{$rpt}};
            my $phone = $drec[1];
            $sth->execute( $phone );   # "execute" repeatedly
            my @dbrec = $sth->fetchrow_array;
            if ( @dbrec ) {
                ...
            }
        }
    }
}