### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

None,
                self.dag_id,
            )
            info = None
        if info is None:
            
            
            if not align:
                yield DagRunInfo.interval(earliest, latest)
            return
        
        
        if not align and info.logical_date != earliest:
            yield DagRunInfo.interval(earliest, info.data_interval.start)
        
        while info is not None:
            yield info
            try:
                info = self.timetable.next_dagrun_info(
                    last_automated_data_interval=info.data_interval,
                    restriction=restriction,
                )
            except Exception:
                self.log.exception(
                    "Failed to fetch run info after data interval %s for DAG %r",
                    info.data_interval if info else "<NONE>",
                    self.dag_id,
                )
                break
    def get_run_dates(self, start_date, end_date=None) -> list: