First, you need to obtain access to the New York Times API to fetch data. Visit the New York Times Developer Network at https://developer.nytimes.com, create an account, and generate an API key for the services you want to access, such as articles, archives, or any other available endpoint.

Use Python to make HTTP requests to the New York Times API. Utilize the `requests` library to send GET requests and retrieve data. For example, you can use the following script to fetch articles:

```python
import requests

api_key = 'your_api_key'
url = 'https://api.nytimes.com/svc/search/v2/articlesearch.json'
params = {
'q': 'your_query',
'api-key': api_key
}

response = requests.get(url, params=params)
data = response.json()
```

Once you have retrieved the data, transform it into a format suitable for insertion into TiDB. This typically involves converting the JSON response into a structured format like CSV or directly into SQL insert statements. Use Python's `pandas` library for handling and transforming data efficiently.

```python
import pandas as pd

articles = data['response']['docs']
df = pd.json_normalize(articles)
df.to_csv('articles.csv', index=False)
```

Ensure that your TiDB environment is ready for data insertion. This involves setting up TiDB on your local machine or in a cloud environment. You can download TiDB from https://pingcap.com/download/ and follow the installation instructions for your operating system. Ensure that you have access credentials and know the hostname, port, username, and password to connect to your TiDB instance.

Before inserting data, create a database and a corresponding table in TiDB where the data will reside. Use a MySQL client or command line client to execute SQL commands:

```sql
CREATE DATABASE nytimes_data;
USE nytimes_data;

CREATE TABLE articles (
id VARCHAR(255) PRIMARY KEY,
headline TEXT,
pub_date DATETIME,
web_url TEXT,
snippet TEXT
);
```

Load the transformed data into TiDB. You can use the TiDB command line tools or a Python script with a MySQL connector to insert data. If using a CSV file, consider using the `LOAD DATA` SQL command:

```sql
LOAD DATA LOCAL INFILE 'articles.csv'
INTO TABLE articles
FIELDS TERMINATED BY ','
ENCLOSED BY '"'
LINES TERMINATED BY '\n'
IGNORE 1 LINES;
```

Alternatively, use Python with `mysql-connector-python`:

```python
import mysql.connector

connection = mysql.connector.connect(
host='your_tidb_host',
user='your_username',
password='your_password',
database='nytimes_data'
)
cursor = connection.cursor()

for index, row in df.iterrows():
cursor.execute("""
INSERT INTO articles (id, headline, pub_date, web_url, snippet)
VALUES (%s, %s, %s, %s, %s)
""", (row['id'], row['headline.main'], row['pub_date'], row['web_url'], row['snippet']))

connection.commit()
cursor.close()
connection.close()
```

After the data load, it's crucial to verify that the data in TiDB matches what was retrieved from the New York Times API. You can do this by running SELECT queries and comparing the result with your local data file or script outputs. Ensure that all records are inserted and that there are no discrepancies or errors during the load process.