while (<FILE>) {   ## (note: using logical indenting)
   if ($found) {
     if (/^$number/) {
       if (/^-/) {
         print COPYFILE "$_\n";
         $found = 0;         ## <<== this is why
       }
     }
   }
   elsif ...
}

##</code><code>##

$found = 0;
while (<FILE>) {
   if ( $found ) {
      if ( /^\d+/ ) {  # does line start with a digit?
         $found = 0;   # then we're done printing stuff
      }
      elsif ( /^-/ ) { # otherwise, if there's stuff
         print COPYFILE;  # then we print it
      }
   }
   elsif ( /^$name/ ) {
      $found = 1;
   }
}