Begin by preparing the data you wish to move from CockroachDB. Identify the tables and columns required and ensure the data is clean and well-structured. Use SQL queries to extract this data, verify its integrity, and format it appropriately.

Use CockroachDB's built-in SQL capabilities to export data. You can use the `COPY` command to export data into CSV files. For example:
```sql
COPY my_table TO '/path/to/export/my_table.csv' WITH CSV;
```
Ensure you have access rights to write to the specified file path and that the data export captures all necessary fields.

Once the CSV files are created, transfer them to Amazon S3. This can be done using AWS CLI or any secure method like `scp` or `rsync`. For AWS CLI, the command would be:
```bash
aws s3 cp /path/to/export/my_table.csv s3://my-bucket/path/to/store/
```
Ensure that your AWS credentials are configured properly and you have permissions to write to the S3 bucket.

Log into your Amazon Redshift cluster and create tables that mirror the structure of your CockroachDB tables. Use SQL commands to define the schema, ensuring data types and constraints are compatible. For example:
```sql
CREATE TABLE my_table (
column1 INT,
column2 VARCHAR(255),
...
);
```

Utilize the `COPY` command in Redshift to load data from the S3 bucket into your Redshift tables. The command should include AWS access credentials and specify formatting options like CSV. For example:
```sql
COPY my_table
FROM 's3://my-bucket/path/to/store/my_table.csv'
IAM_ROLE 'arn:aws:iam::your-account-id:role/RedshiftRole'
DELIMITER ','
IGNOREHEADER 1;
```
Verify that the Redshift IAM role has the necessary permissions to access the S3 bucket.

After loading the data, perform verification checks to ensure data integrity. Run SQL queries to compare row counts, check for null values, and validate key constraints against the original data in CockroachDB. This step ensures that the data migration was successful and accurate.

Post data import, optimize your Redshift cluster for performance. This can involve analyzing the distribution keys and sort keys, updating statistics, and performing vacuum operations to reclaim space and improve query performance. For example:
```sql
VACUUM;
ANALYZE;
```
Regular maintenance and optimization ensure that your queries run efficiently.

By following these steps, you can manually migrate data from CockroachDB to Amazon Redshift without relying on third-party connectors or integrations.