doc: update documentation checks

We should really use the reusable workflows, but this is currently
not possible since we require some non-standard processing.
Therefore, keep the current checks and update them to the latest
framework.

Also, build the docs first and then run the spellcheck on the
docs that are already built (instead of building again).

Signed-off-by: Ruth Fuchss <ruth.fuchss@canonical.com>

.github/workflows/tests.yml

@@ -492,6 +492,12 @@ jobs:
           set -eux
           make doc-lint
 
+      - name: Build docs (Sphinx)
+        run: |
+          set -eux
+          make doc
+          if [ -s doc/.sphinx/warnings.txt ]; then cat doc/.sphinx/warnings.txt; exit 1; fi
+
       - name: Run spell checker
         run: |
           set -eux
@@ -510,18 +516,12 @@ jobs:
           set -eux
           make doc-linkcheck
 
-      - name: Build docs (Sphinx)
-        run: |
-          set -eux
-          make doc
-          if [ -s doc/.sphinx/warnings.txt ]; then cat doc/.sphinx/warnings.txt; exit 1; fi
-
       - name: Upload documentation artifacts
         if: always()
         uses: actions/upload-artifact@v4
         with:
           name: documentation
-          path: doc/html
+          path: doc/_build
 
   snap:
     name: Trigger snap edge build