### 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)

RemovedInAirflow3Warning,
                stacklevel=2,
            )
            dry_run = True
        if recursion_depth:
            warnings.warn(
                "Passing `recursion_depth` to dag.clear() is deprecated.",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
        if max_recursion_depth:
            warnings.warn(
                "Passing `max_recursion_depth` to dag.clear() is deprecated.",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
        state: list[TaskInstanceState] = []
        if only_failed:
            state += [TaskInstanceState.FAILED, TaskInstanceState.UPSTREAM_FAILED]
        if only_running:
            
            state += [TaskInstanceState.RUNNING]
        tis = self._get_task_instances(
            task_ids=task_ids,
            start_date=start_date,
            end_date=end_date,
            run_id=None,
            state=state,
            include_subdags=include_subdags,
            include_parentdag=include_parentdag,