### 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_
children_resources = api_resources.get(resource_id, [])
                if value in children_resources:
                    raise BadRequestException("Resources cannot be cyclical.")
                new_sibling_resources = api_resources.get(value, [])
            else:  
                future_path_part = patch_operation.get("value")
                new_sibling_resources = api_resources.get(moto_resource.parent_id, [])
            for sibling in new_sibling_resources:
                sibling_resource = moto_rest_api.resources[sibling]
                if sibling_resource.path_part == future_path_part:
                    raise ConflictException(
                        f"Another resource with the same parent already has this name: {future_path_part}"
                    )
        
        _patch_api_gateway_entity(moto_resource, patch_operations)
        
        if moto_resource.parent_id != current_parent_id:
            current_sibling_resources = api_resources.get(current_parent_id)
            if current_sibling_resources:
                current_sibling_resources.remove(resource_id)
                
                if not current_sibling_resources:
                    api_resources.pop(current_parent_id)
        
        future_sibling_resources = api_resources[moto_resource.parent_id]
        future_sibling_resources.append(resource_id)
        response = moto_resource.to_dict()
        return response
    
    def get_method(