### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

APiGateway Model that being created by CDK project stores the JSON schema with a single quotes instead of double quotes which leading to an error in parsing this json object. 

you can refere to this article to see the steps 

https://dev.to/dvddpl/reduce-lambda-invocations-and-code-boilerplate-by-adding-a-json-schema-validator-to-your-api-gateway-15af

and it works fine on AWS account 

also you can look on the attached image how localstack stores the JSON object with a single quotes, I tried to edit it manually through UI but it return it back with single quotes

![Screen Shot 2022-10-18 at 4 28 08 PM](https://user-images.githubusercontent.com/106184681/196769273-0834865a-2fb3-4efc-b3a9-9da6b2ad511d.png)

### Expected Behavior

store the json schema in correct JSON format

### How are you starting LocalStack?

With the `localstack` script

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    localstack start

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

    cdklocal deploy


### Environment

```markdown
- OS:OSX
- LocalStack: 1.0.4
```


### Anything else?

_No response_
result = to_response_json("authorizer", data, api_id=api_id)
    result = select_from_typed_dict(Authorizer, result)
    return result
def to_validator_response_json(api_id, data):
    result = to_response_json("validator", data, api_id=api_id)
    result = select_from_typed_dict(RequestValidator, result)
    return result
def to_documentation_part_response_json(api_id, data):
    result = to_response_json("documentationpart", data, api_id=api_id)
    result = select_from_typed_dict(DocumentationPart, result)
    return result
def to_base_mapping_response_json(domain_name, base_path, data):
    self_link = "/domainnames/%s/basepathmappings/%s" % (domain_name, base_path)
    result = to_response_json("basepathmapping", data, self_link=self_link)
    result = select_from_typed_dict(BasePathMapping, result)
    return result
def to_account_response_json(data):
    result = to_response_json("account", data, self_link="/account")
    result = select_from_typed_dict(Account, result)
    return result
def to_vpc_link_response_json(data):
    result = to_response_json("vpclink", data)
    result = select_from_typed_dict(VpcLink, result)
    return result
def to_client_cert_response_json(data):