Compare and contrast:

# use this form of foreach instead of 'my $FH = $_'
foreach my $FH (@files) {
    
    # use 3-arg form of open() for safety, and deal with
    # being unable to open the file.
    open my $FILE, '<', $FH or die("Cannot read $FH: $!");

    # deal with each line of the file.  while loop exits
    # when EOF is reached.
    my $batch;
    OUTER: while (<FILE>) {
        $batch .= $_; # cat this line to $batch

        # if we just read a line that's 'GO', run the batch and
        # reset the variable $batch to be empty
        if (m/^\s*go\s*$/i) {
            $dbh->do($batch);
            # you should check for and deal with DBI errors here
            $batch = '';
        }
    }

    # if we didn't reach a 'GO', there will be stuff in $batch, so
    # we should report that fact.
    warn "File '$FH' contained unexecuted commands:\n$batch\n";
}
[download]

Once you get that down, it becomes much easier to explore other algorithms, like this one:

foreach my $FH (@files) {
    open my $FILE, '<', $FH or die("Cannot read $FH: $!");
    my $data = join('', <$FILE>); # slurp file!

    # split on lines containing only 'GO' (and optionally, whitespace)
    foreach my $batch ( split /\n\s*go\s*\n/i, $data ) {
        # if you need the 'go', you'll need to do $batch.="\nGO"; here
        $dbh->do($batch);
        # error handling here
    }
}
[download]