I thought it was okay to use multiple nested instances of $_, and that each was local to its block, but clearly not...

That's true for foreach, but not while (<>):

use warnings;
use strict;
use Data::Dump;

$_ = "foo";
for ("x","y") {
    dd $_;
}
dd $_;
while (<DATA>) {
    dd $_;
}
dd $_;

__DATA__
hello
[download]

Output:

"x"
"y"
"foo"
"hello\n"
undef
[download]