Improve short codes live tests

[ericasp16]

Reorganize and improve the short codes live tests to be more robust and thoroughly check each endpoint. Improve logging/error checking/validation, etc

[ramya-rao-a]

@ericasp16

Unrelated to this PR, but can you please add an entry to the codeowners file for this packege? See https://github.com/Azure/azure-sdk-for-js/blob/main/.github/CODEOWNERS#L36-L42. This will ensure that your PRs get labelled properly and get the right reviewers get looped in automatically

[ramya-rao-a]

Another unrelated note to this PR, but related to this package is at #18688
Please take a look at that separately as well.

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