### 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)

warnings.warn(
            "`DAG.normalize_schedule()` is deprecated.",
            category=RemovedInAirflow3Warning,
            stacklevel=2,
        )
        with warnings.catch_warnings():
            warnings.simplefilter("ignore", RemovedInAirflow3Warning)
            following = self.following_schedule(dttm)
        if not following:  
            return dttm
        with warnings.catch_warnings():
            warnings.simplefilter("ignore", RemovedInAirflow3Warning)
            previous_of_following = self.previous_schedule(following)
        if previous_of_following != dttm:
            return following
        return dttm
    @provide_session
    def get_last_dagrun(self, session=NEW_SESSION, include_externally_triggered=False):
        return get_last_dagrun(
            self.dag_id, session=session, include_externally_triggered=include_externally_triggered
        )
    @provide_session
    def has_dag_runs(self, session=NEW_SESSION, include_externally_triggered=True) -> bool: