1. Add my before $paragraph in while(defined($paragraph=$enumerate->Next())).
2. Typo: Change ListParagraph to ListParagraphs.
3. Typo: Change ListFormatListLevelNumber to ListFormat->ListLevelNumber.
4. You might need ListFormat->ListValue instead of (or in addition to) ListFormat->ListLevelNumber.
5. To match your VB code, check for $count->{Count} == 1 inside the if (defined $count) {...} block.

Working, tested code:

#!/usr/bin/perl
use strict;
use warnings;
use Cwd qw( abs_path );
use Data::Dumper;

doc2pt( "C:/a/perls/pm/PM_932635_data.doc" );

sub doc2pt {
    die if @_ != 1;
    my ( $doc_path ) = @_;
    
    ( my $out_path = $doc_path ) =~ s{\.doc$}{.out}
        or die "doc_path '$doc_path' does not end in '.doc'";

    require Win32::OLE;
    require Win32::OLE::Enum;

    # NOTE: Win32::OLE appears to need abs path
    my $abs_doc_path = abs_path($doc_path);

    my $document = Win32::OLE->GetObject($abs_doc_path)
        or die "Can't GetObject($abs_doc_path) $!\n";
    print "Extracting Text ...\n";
    open my $out_fh, '>', $out_path
        or die "Can't open output file '$out_path': $!";

    my $debug = sub {
        die if @_ != 2;
        my ( $name, $value ) = @_;
        local $Data::Dumper::Useqq = 1;
        local $Data::Dumper::Terse = 1;
        printf {$out_fh} "%-10s ==> %s", $name, Dumper $value;
    };

    my $paragraphs = $document->Paragraphs();
    my $enumerate = Win32::OLE::Enum->new($paragraphs);
    while ( my $paragraph = $enumerate->Next() ) {
        my $style = $paragraph->{Style}->{NameLocal};
        $debug->( style => $style );
        my $range = $paragraph->{Range};
        my $count = $range->ListParagraphs();
        if ( defined $count ) {
            my $real_count = $count->{Count};
            my $paranum = $range->ListFormat->ListLevelNumber();
            my $paraval = $range->ListFormat->ListValue();
            $debug->( real_count => $real_count );
            $debug->( paranum    => $paranum );
            $debug->( paraval    => $paraval );
        }
        my $text = $paragraph->{Range}->{Text};
        $text =~ tr{\n\r}{}d;
        $text =~ tr{\x0b}{\n};
        $debug->( text => $text );
    }
    close $out_fh;
}
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