"I already got this part, i.e.: logged in and got auth url"

This isn't in the code you've posted, it doesn't match the example from the docs.

"I can't figure out how to authenticate and ultimately, list posts. I would really appreciate a working code example."

With the exception of the Instagram documentation that I looked at out of curiosity, I know nothing about their system, what it does or why anyone would use it. That said the API::Instagram documentation has examples of how to get a users 'feed' (if this is what you mean by posts?):

my $feed    = $my_user->feed( count => 5 );
 
for my $media ( @$feed ) {
 
        printf "Caption: %s\n", $media->caption;
        printf "Posted by %s at %s (%d likes)\n\n", $media->user->user
+name, $media->created_time, $media->likes;
 
}
[download]

"I would really appreciate a working code example."

I don't have any use case for this, and am not going to jump through the hoops to get an account/API keys etc.. My suggestion would be to Use the source Luke. You have the API documentation from Instagram, you have the Perl source code for this module. Work the problem. Perhaps some of the tests can point you in the right direction. Once you work out what you want, consider suggesting improvements to the module documentation if you feel this would be of benefit.