# line mode
...

{
   # paragraph mode
   local $/ = '';
   ...
}

# line mode
...
[download]

Your case:

sub process_csv {
   ...

   open(my $fh, '<', ...)
      or die(...);

   while (<$fh>) {
      ...
      process_file(...);
      ...
   }

   ...
}

sub process_file {
   ...

   local $/ = '';

   open(my $fh, '<', ...)
      or die(...);

   while (<$fh>) {
      ...
   }

   ...
}
[download]

Beware. If proces_file calls functions, paragraph mode will still be active. That means don't call process_csv (directly or indirectly) from process_file without fixing $/.

I think local would be your best friend here. You probably can do what you need by defining well contained subroutines which affect the line separator in a minimum scope and then your programming will switch transparently between line and paragraph modes.

Something like this

sub read_line {
    my $h = shift;
    local $/ = "\n"; 
    return <$h>
}

sub read_para {
    my $h = shift;
    local $/ = "";
    return <$h>;
}

use Fatal qw(open);
open H, '<', 'files.csv';
while (my $txt = read_file(*H)) {
      # process the CSV line and make the filename available in $fn
      open TXT, '<', $fn;
      my @paragraphs = read_para(*TXT);
      # process
}
[download]

Update: ikegami (at Re^2: Using both line mode and paragraph mode in the same program) pointed how "\n\n" is the wrong way to get paragraph mode (as can be read in the docs he mentioned). So I got it right with "".

Paragraph mode is "". "\n\n" is different. See $/