How to gracefully shutdown pods when background service is running in Kubernetes cluster?

Question

I have a .Net application hosted in Azure Kubernetes cluster. One is Web application - TestWeb. Another one is console library running as background service - TestWorker. TestWeb receives the request from user and pushes the message to Storage Queue st1, whereas TestWorker reads the message from st1 and processes it. Suppose the deployment is created in cluster AKS1 for both service. Now I want to create new cluster AKS2 and deploy the services TestWeb and TestWorker writing to and reading from same queue st1. I can update the traffic manager endpoint to AKS2 after deployment in AKS2 succeeds. In this way, all the request will route to TestWeb hosted in AKS2. However, TestWorker in AKS1 and AKS2 will still be reading from same queue. How can I gracefully shutdown pods in AKS1 so that TestWorker deployed in AKS2 only reads from storage queue st1? I can set replica count to 0 for all pods in AKS1 but this will not ensure that messages picked up from the queue by TestWorker in AKS1 is completely processed. If I set grace period to more than 30s , It may be possible that message is read at last nth second and pod shutdowns abruptly. What is the best way to gracefully shut down the pod in this case?