📬 Issue #, if available:
n/a

✍️ Description of changes:
#982 broke our integration test since it did not enable the new graceful-shutdown feature flag but used the new spawn_graceful_shutdown_handler(). Sorry about that, I missed fixing it when i removed graceful-shutdown from default features partway through the PR.

Sample failure: https://github.com/awslabs/aws-lambda-rust-runtime/actions/runs/14901777878/job/41855115960

I also included a github CI update to build the integ test crate on new PRs. That way we can catch this kind of regression prior to merge.

🔏 By submitting this pull request

• [x ] I confirm that I've ran cargo +nightly fmt.
• [ x] I confirm that I've ran cargo clippy --fix.
• [x ] I confirm that I've made a best effort attempt to update all relevant documentation.
• [x ] I confirm that my contribution is made under the terms of the Apache 2.0 license.