react-native async function returns promise but not my json data?

Another approach similar to the original code of the questioner:

async componentDidMount() {
    let data = await this.getData();
    console.log(data);    
    this.setState({
      dataSource:this.state.dataSource.cloneWithRows(data),
    })  
  }

Since getData() is a promise, you should be able to obtain the data in a then block as follows:

componentDidMount() {
  this.getData()
    .then((data) => {
      this.setState({
        dataSource:this.state.dataSource.cloneWithRows(data),
      })  
    });
}