Fix all github actions issues + depreciations

[muellerzr]

Fixes a duplicate of writing to the same cache location that was causing a partial slowdown. Also goes ahead and changes set-output to $GITHUB_OUTPUT as this will be githubs new way of modifying bits like that safer.

cc @ydshieh so you're aware of it, docs are here: https://docs.github.com/en/actions/using-workflows/workflow-commands-for-github-actions#environment-files

[HuggingFaceDocBuilderDev]

The documentation is not available anymore as the PR was closed or merged.

[ydshieh]

Thank you @muellerzr for updating my knowledge!

[sgugger]

Thanks!