Create new clientsession for NYT Games

[joostlek]

Proposed change

Create new clientsession for NYT Games because of cookies

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Deprecation (breaking change to happen in the future)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes NYT Games only allows one account #127377
• This PR is related to issue:
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
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• The code has been formatted using Ruff (ruff format homeassistant tests)
• Tests have been added to verify that the new code works.

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  Updated by running python3 -m script.gen_requirements_all.
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