This might work:

my $timeout = 10; # 10 seconds timeout
SOURCE:
foreach my $source (@sources) {
    my dbh = connect_to_db($source);
    my sth = $dbh->prepare( $your_query );
    my $eval_result = eval {
        local $SIG{ALRM} = sub { die "alarm\n" };
        alarm( $timeout );
        $sth->execute( $your_params );
        alarm( 0 );
        1;
    };
    
    if ( $eval_result ) {
        last SOURCE;
        # ALL OK
    }
    else {
        print "Timeout happens!\n";
    }
}
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It might be useful to see also How to timeout if a call to an external program takes too long and Re: Question on "Effective Perl Programming" (;1 > $@).

Note that the above will actually change $@ when it's run, even though it doesn't use it otherwise. If you really want code that doesn't touch $@, using local is the way to go.