This may work (UNTESTED):

#!/usr/bin/perl
use strict;
use warnings;

use File::Find;

@ARGV == 3 or die "usage: $0 DIR OLD_EXT NEW_EXT\n";

my $dir = $ARGV[ 0 ];
my $old = $ARGV[ 1 ];
my $new = $ARGV[ 2 ];

my @list;
find sub {
    push @list, $File::Find::name if -f && /\.\Q$old\E\z/;
    }, $dir;

foreach my $old_name ( @list ) {
    my $new_name = $old_name =~ s/\.\Q$old\E\z/.$new/r;
    rename $old_name, $new_name or die "Cannot rename '$old_name' beca
+use: $!";
    }
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You could use the -d test to look for entries that are themselves directories and then recurse down into them but at that point you'd have started down the path of reimplementing what something like Path::Tiny (or File::Find, or File::Find::Rule, or Path::Iterator::Rule, or . . .) could do for you off the shelf so you'd be better of using that to begin with.

## presuming your sample vars for dir, old, new
## Somewhat untested, but . . .
use Path::Tiny qw( path );

my $iter = path( $dir )->iterator( { recurse => 1 } );

while( my $path = $iter->() ) {
  next unless $path->is_file and $path =~ m{\. $old $}x;

  my $new_path = $path->basename( $old ) . ".$new";

  unless( my $ret = $path->move( $new_path ) ) {
    warn "Problem moving '$path': $ret\n";
  }
}
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See the FAQ: How do I traverse a directory tree?