This doesn't answer to your original problem of reading a file with multiple while loops, but it does present another WTDI that, to me, seems much more natural.
Since your file is really small, read it in its entirety into and array of lines, a la:
my $filename = 'filename.txt';
open my $ifh, '<', $filename;
my @lines = <$ifh>;
Now you can process the content of you file to your heart's content by simply maintaining the relevant indices for the lines you want to be dealing with, and iterating via "for" loops. You can do this as many times as you want without having to worry about iterating via <>...
Updated: Added blurb re: for-loop iteration
What can be asserted without proof can be dismissed without proof. - Christopher Hitchens