The TTLs (Time to live) of a directory entry is not advanced when the upstream server can not be reached. Further, is an entry is hit, its ttl is reset so that it will not be purged.
But modern dns servers never purge anymore, it takes more time than its worth. Which is why domain names often work even if you don't pay your registration. Not all registrars will take the time to null-route you the instant your bill goes unpaid.
As for your comment "the signal will basically have to hunt for the website", thats nonsense.
There is no hunting involved. Your machine asks your ISP what the IP for www.foxnews.com is and it gets a number handed back. If for some unlikely reason your ISP didn't have it, it in turn asks its upstream provider. If that provider does not have it goes higher. As soon as this request reaches a server that claims to be authoritative it will go no higher. Most requests (something like 99.999999%) are handled by your ISP and never make it any higher than that.
Once the IP is known, the packets are routed by the router. Again, there is no hunting. Your ISP maybe has two possible routes out if its office. The router chooses one, and sends it off. The next router picks the best route it has and does the same. There is no hunting....
But my given understanding is that a great many sites have
relatively very short TTLs set for their severs and
intranets, and that in many cases, even a matter of a
25 hrs, can cuse a gret many of them to expire.