CI changelogs: switch to XDG, fix cache key (backport #9849)

[mergify]

This is the same as #9845 but for the changelogs.yml workflow.

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This is an automatic backport of pull request #9849 done by [Mergify](https://mergify.com).

[mergify]

Cherry-pick of af5d606 has failed:

On branch mergify/bp/3.12/pr-9849
Your branch is up to date with 'origin/3.12'.

You are currently cherry-picking commit af5d60613.
  (fix conflicts and run "git cherry-pick --continue")
  (use "git cherry-pick --skip" to skip this patch)
  (use "git cherry-pick --abort" to cancel the cherry-pick operation)

Unmerged paths:
  (use "git add <file>..." to mark resolution)
	both modified:   .github/workflows/changelogs.yml

no changes added to commit (use "git add" and/or "git commit -a")

To fix up this pull request, you can check it out locally. See documentation: https://docs.github.com/en/pull-requests/collaborating-with-pull-requests/reviewing-changes-in-pull-requests/checking-out-pull-requests-locally

[ulysses4ever]

I unlabel it for merging while we're investigating the fix-whitespace failure in #10007

[ulysses4ever]

All right, back on track. Some analysis is in #9818 (comment)