GitHub Actions: Deprecating save-state and set-output commands

[roykho]

On May 31st GitHub is deprecating save-state and set-output commands which will impact some of our current actions workflow. Our workflows will start to fail. Ref. https://github.blog/changelog/2022-10-11-github-actions-deprecating-save-state-and-set-output-commands/

This issue is to track the change over to using "{name}={value}" >> $GITHUB_STATE and "{name}={value}" >> $GITHUB_OUTPUT syntax. So far I have gathered several workflows that will require this change. This is based on the annotation of the actions output (for example https://github.com/woocommerce/woocommerce-blocks/actions/runs/4722359005). It is possible some of these may be used from an upstream dependency and if that's the case we still need to make sure we either contact the upstream provider if we're not the maintainers OR find an alternative dependency.

• Check Modified Assets
• Deploy to WordPress.org
• Monitor TypeScript errors

This task can be a PR or at the very least a comment describing the investigation made and what dependencies are actually causing this warning with deprecated commands.