I understand you are looking for an out-of-box App Service solution for Disaster Recovery (DR).
Just to highlight, the recommendations for achieving resiliency and availability, which broadly falls into one of two categories (healthy items and risk items). So, depending on the requirement, there are multiple approaches involving using additional service.
Azure App Service can be deployed across availability zones (AZ) to help you achieve resiliency and reliability for your business-critical workloads. This architecture is also known as zone redundancy.
To prepare for disaster recovery in a multi-region geography, you may use either an active-active or active-passive architecture, involved using Azure Front Door service (instead of Traffic Manager | I understand you are looking to avoid integrating with additional service).
I suggest you take a look at the outlined doc Reliability in Azure App Service for details on the disaster recovery guidance. Plus, doc section Back up & restore vs. disaster recovery ( review the requirement, and recommended options)
Also, see Tutorial: Create a highly available multi-region app in Azure App Service
Hope this helps! Kindly let us know.
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