### 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_
self, context: RequestContext, request: TestInvokeMethodRequest
    ) -> MethodResponse:
        
        
        raise NotImplementedAvoidFallbackError
    
    @handler("CreateAuthorizer", expand=False)
    def create_authorizer(
        self, context: RequestContext, request: CreateAuthorizerRequest
    ) -> Authorizer:
        
        api_id = request["restApiId"]
        store = get_apigateway_store(account_id=context.account_id, region=context.region)
        if api_id not in store.rest_apis:
            
            
            raise ConflictException(
                "Unable to complete operation due to concurrent modification. Please try again later."
            )
        authorizer_id = short_uid()[:6]  
        authorizer = deepcopy(select_from_typed_dict(Authorizer, request))
        authorizer["id"] = authorizer_id
        authorizer["authorizerResultTtlInSeconds"] = int(
            authorizer.get("authorizerResultTtlInSeconds", 300)
        )
        store.rest_apis[api_id].authorizers[authorizer_id] = authorizer
        response = to_authorizer_response_json(api_id, authorizer)
        return response
    def get_authorizers(