### 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_
if len(request_parameters_names) != len(request_parameters):
                raise BadRequestException(
                    "Parameter names must be unique across querystring, header and path"
                )
        need_authorizer_id = authorization_type in ("CUSTOM", "COGNITO_USER_POOLS")
        store = get_apigateway_store(account_id=context.account_id, region=context.region)
        rest_api_container = store.rest_apis[rest_api_id]
        if need_authorizer_id and (
            not authorizer_id or authorizer_id not in rest_api_container.authorizers
        ):
            
            raise BadRequestException(
                "Invalid authorizer ID specified. "
                "Setting the authorization type to CUSTOM or COGNITO_USER_POOLS requires a valid authorizer."
            )
        if request_validator_id and request_validator_id not in rest_api_container.validators:
            raise BadRequestException("Invalid Request Validator identifier specified")
        if request_models:
            for content_type, model_name in request_models.items():
                
                if model_name == EMPTY_MODEL:
                    continue
                if model_name not in rest_api_container.models:
                    raise BadRequestException(f"Invalid model identifier specified: {model_name}")
        response: Method = call_moto(context)
        remove_empty_attributes_from_method(response)
        moto_http_method = moto_resource.resource_methods[http_method]
        moto_http_method.authorization_type = moto_http_method.authorization_type.upper()