Hello! I will tell you a couple of things that might help you to take a good decision.
-So first of all, you can just apply if you are within 90 days of your refusal notice
- meantime you wait for the answer you are allow to be in the country, so I wish you luck to get the answer before you guys leave. You don't have any status but you are legal in the country
- you need to know the reasons why you got your application denied. So in your letter you need to prove you don't want to stay longer, or forever. In my case I made a good letter, I showed the contract which said the day until I paid the apartment, the money I have in Canadian accounts+ mine own from my country and some good reasons why I'm not planning on staying forever there. And last but not least, I didn't ask for 3-4 months I wrote "20 of June" to prepare my departure and my place was rented til that point.
If you decide to stay without doing anything probably nothing will happen because they won't be looking for you, they don't have that time . But you risk when you try to come back, that they ask you some questions or the possibility of not letting you in.