I ended up using:

$rest = $messagebody;
@text=();
while($rest ne '') {
    $rest =~ /(.{1,140}\W)/ms;
    push @text, $1;
    $rest = $';
}

Text::Wrap doesn't provide the desired behavior, unless I were to do a tr/// to translate all the newlines into some strange character, do the wrap, and then tr the strange characters back to newlines. This way seemed a little cleaner.