One aspect of Network Solutions business is managing some of the root zones. Those are the servers that ultimately tell the internet what other servers to go look at to resolve queries. These are the "root" nameservers. So while Network Solutions is in charge of the root zones themselves, and does run a bunch of root nameservers, there are other root nameservers run by Various organizations and collectively they can handle a massive amount of work. That is not what is being attacked. If somone DID successfully DDOS a large enough portion of the root nameserver infrastructure, the internet would largely grind to a halt. It is designed suitably sized, and run by large enough players that this is not likely.

The DNS service that is being attacked is the one they offer end users, usually along with domain registration. There is no particular reason you need to use their services when registering a domain with them, although, like most rigistrars, their website will make this deliberately confusing, and may require you to use their DNS service if you also want to use some of their other value-added services along with your registration (like email forwarding, websites, stuff like that).

Unless you are using other services from NetSol that require you to use their DNS, you are free to use whatever DNS infrastructure you like, including your own.

(If DDOS is a business concern, I'd go with someone big who deliberately does just DNS.. Neustar or one of those)