Identify and gain access to the data you need from the New York Times. If the data is available via their API, register for an API key through their developer portal. If the data is not available via an API, check for public datasets or consider using web scraping techniques, ensuring compliance with their terms of service.

Use Python to extract the data. For API access, utilize the `requests` library to make HTTP requests to the New York Times API and retrieve the data in JSON format. If web scraping is necessary, use libraries like `BeautifulSoup` or `Scrapy` to extract the required information from HTML pages.

Once the data is extracted, process and transform it into a CSV format which is suitable for loading into Redshift. Use Python’s `pandas` library to manipulate the data, clean it (e.g., handle missing values), and export it to a CSV file using `DataFrame.to_csv()` method.

Log in to your AWS account and create an S3 bucket where the CSV file will be temporarily stored. This bucket acts as an intermediary storage location because Redshift can load data from S3 directly. Ensure proper permissions are set for the bucket to allow Redshift access.

Use the AWS CLI or Python’s `boto3` library to upload the CSV file to the S3 bucket. For AWS CLI, use the command `aws s3 cp yourfile.csv s3://your-bucket-name/`. For Python, establish a session using `boto3`, and use the `upload_file()` method to upload your file.

Ensure your Redshift cluster is running and accessible. Establish a connection using SQL clients like SQL Workbench/J. Create a table in your Redshift cluster that matches the schema of your CSV data. Use SQL `CREATE TABLE` statements to set up the structure.

Execute a `COPY` command in Redshift to load data from the S3 bucket into your Redshift table. The basic syntax is:
```sql
COPY your_table_name
FROM 's3://your-bucket-name/yourfile.csv'
IAM_ROLE 'your-iam-role-arn'
CSV;
```
Ensure your IAM Role has the necessary permissions to access S3. This command will transfer the data from S3 into your Redshift table efficiently.

By following these steps, you can move data from the New York Times to a Redshift destination without relying on any third-party connectors or integrations. Always make sure to adhere to data privacy laws and terms of service when handling data.