[ie/newgrounds] Fix login and clean up extraction

[Grub4K]

IMPORTANT: PRs without the template will be CLOSED

Description of your pull request and other information

Based on #9334 by @mrmedieval, thx for the groundwork

• Fixes [Newgrounds] Error 401, Adult-rated video NSFW #9324
• Fixes [newgrounds] Cookies causes video to download profile page #6488

Template

Before submitting a pull request make sure you have:

• At least skimmed through contributing guidelines including yt-dlp coding conventions
• Searched the bugtracker for similar pull requests
• Checked the code with flake8 and ran relevant tests

In order to be accepted and merged into yt-dlp each piece of code must be in public domain or released under Unlicense. Check all of the following options that apply:

• I am the original author of this code and I am willing to release it under Unlicense
• I am not the original author of this code but it is in public domain or released under Unlicense
  See Add login to Newgrounds extractor #9334

What is the purpose of your pull request?

• Fix or improvement to an extractor (Make sure to add/update tests)
• New extractor (Piracy websites will not be accepted)
• Core bug fix/improvement
• New feature (It is strongly recommended to open an issue first)