Maybe what you need to do is move the "exit" command for sqlplus out of the "$script_name" file and into a print statement:

...
print SQLP "\@$script_name; \n" or die "cannot write in doScript"; 
# add:
print SQLP "exit\n" or die "cannot write exit to sqlplus in doScript";
close SQLP or die "cannot close in doScript"; 

...

****** Start  t.sql 
  select sysdate from dual 
  / 
****** end t.sql
...
[download]

Note: that's just something to try -- I don't have a clue whether it will work.

Is it the case that you have no hope of installing DBI and DBD::Oracle on the platforms involved? This would make it a lot easier to communicate with oracle through perl, without exposing account names/passwords.

One other point: are you sure you want to redirect sqlplus output to $output_file_name via the ">>" (append) operator? Do you intend to read that file back after a sequence of queries has been completed, or is it being read back after each query?