First of all, at GoDaddy, you would need to register the private nameservers you are using. You can put them both onto the same IP. In your Domain Manager at GoDaddy, click on the domain, then scroll to the bottom and look at the far left hand corner. There will be a Hosts Summary (add) area. Click the add link. Enter ns1 in the first field, then the IP for the nameserver in the "Host IP 1" field. Click OK. Wait around 3-5 minutes and then add the second nameserver as ns2 with the same IP (it takes GoDaddy around that long to let you add a second one).
After you have registered the private nameservers there, then change the domain's nameservers to those new ones in Nameservers > Set Nameservers section.
After you've done the above, in cPanel, go to WHM > Basic cPanel/WHM Setup area and put your nameservers there. Click "Add an A entry for this nameserver" and enter the IP if it isn't the correct one, then click the "Add Entry" button. Do this for both nameservers.
At that point, you should have working nameservers provided the private registration at GoDaddy has already propagated.
Of note, the WHM > Nameserver IPs area doesn't do anything in regards to setting the A record or getting them working in cPanel. You don't even need to bother with using that area.
Finally, in regards to your /etc/resolv.conf file, your IPs should not be in that file and, yes, that file does matter. Your resolver determines if your machine can resolve off-server domains, which it needs to do in order for email to function for one thing. The resolver is an off server location to resolve domains, so it isn't supposed to have your IPs in it. You could use a good public resolver instead in that file such as the 4.2.2.2 ones (I believe it's Level3 or VeriSign) or Google's resolvers:
Level3
nameserver 4.2.2.2
nameserver 4.2.2.3
Google
nameserver 8.8.8.8
nameserver 8.8.4.4
I remember back when I had to try to figure this stuff out myself, so I know how complicated it can be the first time to get everything working.