Begin by connecting to your Amazon Redshift cluster using a SQL client tool, such as `psql`, or a Python script with a library like `psycopg2`. You'll need the cluster endpoint, database name, username, and password for authentication.

Use the `UNLOAD` command to export data from Redshift to an Amazon S3 bucket in CSV format. The `UNLOAD` command allows you to specify a query to export data. Ensure your Redshift cluster has the proper IAM role with S3 write permissions.

```sql
UNLOAD ('SELECT * FROM your_table')
TO 's3://your-bucket-name/data/export_'
IAM_ROLE 'arn:aws:iam::your-account-id:role/your-role'
DELIMITER ','
ALLOWOVERWRITE
PARALLEL OFF;
```

Once the data is in S3, use AWS CLI to download the CSV file to your local machine. Ensure you have the AWS CLI installed and configured with access to your S3 bucket.

```bash
aws s3 cp s3://your-bucket-name/data/export_0000.csv ./local_export.csv
```

Use a scripting language like Python to read the CSV file and convert it to JSON format. Python's `csv` and `json` libraries are ideal for this task.

```python
import csv
import json

csv_file_path = './local_export.csv'
json_file_path = './local_data.json'

with open(csv_file_path, mode='r') as csv_file:
csv_reader = csv.DictReader(csv_file)
data = [row for row in csv_reader]

with open(json_file_path, mode='w') as json_file:
json.dump(data, json_file, indent=4)
```

After conversion, validate the JSON file to ensure the data integrity. You can use a JSON validator or a simple script to check for common issues like malformed JSON.

```python
import json

try:
with open(json_file_path, 'r') as json_file:
json.load(json_file)
print("JSON data is valid.")
except json.JSONDecodeError as e:
print(f"Invalid JSON data: {e}")
```

Remove any temporary files generated during the process, such as the CSV file, to maintain an organized workspace.

```bash
rm ./local_export.csv
```

Ensure the local JSON file is stored securely. Set appropriate file permissions and, if needed, encrypt the file to protect sensitive data.

```bash
chmod 600 ./local_data.json
```

Following these steps will allow you to efficiently move data from Amazon Redshift to a local JSON file without relying on third-party connectors or integrations.