Is it possible to connect your existing website with Short.cm? No, it is not.
To connect with Short.cm, you must have a dedicated domain or sub-domain as your branded one for URL shortening purpose.
The most important recommendation is to create your domain for URL shortening. It would be perfect if it fits your brand.
You should remember, when you set up your DNS to remit to our servers, we make your "website hosting". But, actually we are hosting the short links only, and we don't propose web hosting services.
Note: be attentive when you are working on computer where your website is located, and operate with our servers instead. It is paramount. If you try to connect URL for shortening (by changing your DNS settings) to a live website domain, it will certainly crash your website down.