Upgrade react-native-async-storage

[steffenkleinle]

Describe the Problem

react-native-async-storage has a new major release (v3.0.0) with API changes that were out of scope for #3951.

Describe Your Preferred Solution

Upgrade react-native-async-storage and use the new API.
See the migration guide: https://react-native-async-storage.github.io/3.0/migration-to-3/

Describe Possible Alternatives

N/A

Related Issues

#3951

Additional Information

https://react-native-async-storage.github.io/3.0/migration-to-3/