But then you still get warnings?   What I tested with was:

#!/usr/bin/perl -w

use strict;
use warnings;

my  $var_one;
    $var_one = 'now defined';

sub copy_file {
  while( my $x = <DATA>) {
      $x =~ s|(\$\w+)|$1|eeg; 
      print $x;
  };
}

copy_file();

__DATA__
The variable is $var_one.
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As coded above I get the output

The variable is now defined.

If I comment out the   $var_one =   I see

Use of uninitialized value in substitution iterator at anon02.pl line 12, <DATA> line 1.
The variable is .

Any chance the variable name is misspelled in the text file?   When I tried the following data:

__DATA__
The variable is $var_one.
The variable is $var_one2.
The variable is $Var_one.

I got error messages:

The variable is now defined.
Use of uninitialized value in substitution iterator at anon02.pl line 14, <DATA> line 2.
The variable is .
Use of uninitialized value in substitution iterator at anon02.pl line 14, <DATA> line 3.
The variable is .