I am torn between z test and t test. I am leaning towards z test because by the Central Limit Theorem $102>30$ and $69>30$. Am I wrong in my assertion? What do I need to look out for in deciding if to use a z test or t test aside when $\sigma$ is known or not.
Definitely use a t-test. Since you are using a $cohort$ of individuals with the disease (i.e., a sample) you should use the t-distribution. Only use a z-test (Normal distribution) when you are testing something on a population (which in practice isn't super often).