From a quick search on Stack Overflow I saw people suggesting the following way of checking if a cookie exists:
HttpContext.Current.Response.Cookies
You need to use HttpContext.Current.Request.Cookies, not Response.Cookies.
Side note: cookies are copied to Request on Response.Cookies.Add, which makes check on either of them to behave the same for newly added cookies. But incoming cookies are never reflected in Response.
This behavior is documented in HttpResponse.Cookies property:
After you add a cookie by using the HttpResponse.Cookies collection, the cookie is immediately available in the HttpRequest.Cookies collection, even if the response has not been sent to the client.