Yes, it is possible to have your own nameservers
All you need to do is to create nameservers for your domain and point it to our DNS cluster IPs.
dns1.coolicehost.com (132.226.216.127)
dns2.coolicehost.com (51.79.9.7)
Example:
ns1.yourdomain.com >132.226.216.127
ns2.your domain.com > 51.79.9.7
Note: The IPs in these examples could not refer to your case. You need to look at your New Hosting Account email to see the exact IPs.
Then edit nameservers in the DNS editor in the domain in your control panel.
And create A records for ns1 / ns2 yourdomain.com.
You are done.