Anyone know about domain nameservers with dedicated servers?

I’m setting up a new site for a client on their existing dedicated server. On their dedicated server they have a main site and then 2 additional sites that are added as “addon domains” in cPanel. I have the site set up, but now I’m having trouble getting the domain name to propogate. Normally this is a easy process, but I think it has something to do with the fact it’s a dedicated server.

The main domain is registered with Network solutions, and has private name servers set up with Network solutions that are pointing to the dedicated server. The new domain that I’m trying to use for the new site is registered with GoDaddy. Can I use the name servers that are set up with Network Solutions for this new domain with GoDaddy? Or do I need to set up new nameservers with GoDaddy that point to the dedicated server?

Any help will be greatly appreciated!