No need to seek through the file. You already know that what you want to get rid of, if it’s there at all, is at the very end.

#! usr/bin/perl
use strict;
use warnings;

# length of end-of-line marker
my $eol_len = length $/;

opendir my $dh, "." or die "Can't open current directory: $!\n";

while( my ( $file ) = readdir $dh ) {
    next if $file !~ /\.txt/;

    open my $fh, '+<', $file
        or warn( "Can't open $file for read/write: $!\n" ), next;

    # read data from last possible location of end-of-line
    seek $fh, -$eol_len, 2;
    my $tail = <$fh>;

    # if chomp finds an EOL, rewind and truncate
    if( chomp $tail ) {
        seek $fh, -$eol_len, 2;
        truncate $fh, tell $fh;
    }
}
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Makeshifts last the longest.