1. Update cookies after all successful RPCs. Ensures that client adapts to changes in cluster configuration.
2. Reorganized client code to extract more snippets that could be copy-pasted.
3. Docker build now uses the same bazel build as outside code.
Closes#35799
COPYBARA_INTEGRATE_REVIEW=https://github.com/grpc/grpc/pull/35799 from eugeneo:tasks/ssa-example-bug 801205a532
PiperOrigin-RevId: 603805785
Yash Tibrewal