### 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)

if end_date or not self.allow_future_exec_dates:
            end_date = end_date or timezone.utcnow()
            tis = tis.where(DagRun.execution_date <= end_date)
        if state:
            if isinstance(state, (str, TaskInstanceState)):
                tis = tis.where(TaskInstance.state == state)
            elif len(state) == 1:
                tis = tis.where(TaskInstance.state == state[0])
            else:
                
                if None in state:
                    if all(x is None for x in state):
                        tis = tis.where(TaskInstance.state.is_(None))
                    else:
                        not_none_state = [s for s in state if s]
                        tis = tis.where(
                            or_(TaskInstance.state.in_(not_none_state), TaskInstance.state.is_(None))
                        )
                else:
                    tis = tis.where(TaskInstance.state.in_(state))
        
        if include_parentdag and self.parent_dag is not None:
            if visited_external_tis is None:
                visited_external_tis = set()
            p_dag = self.parent_dag.partial_subset(
                task_ids_or_regex=r"^{}$".format(self.dag_id.split(".")[1]),
                include_upstream=False,
                include_downstream=True,
            )