I find it hard to believe that Godaddy is not helping you. I have been with them for years with hosting and domains and have never had a problem with them.
Hello AMessler it was hard for me too. And my new host did propagate my private nameservers days before I asked Godaddy to help me.
The problem clearly was with Godaddy and it is fixed now.
I wrote to the president office at Godaddy and explained them that after 8 years and closely having 200 domains in 3 accounts with them I am leaving.
I did write them about the way their support handled my request with clearly copy and paste all their responses.
What did surprise me that they phoned me twice to Europe and the second time I received apologies for the way Godaddy support have reacted to me and my partner.
And yes it has been outraged that telling me how to enter into my controlpanel instead looking more deeply into the problem.
But I must say that I am pleasantly surprised with this apology from them and also their financial offer (gift) they gave me.
We stopped for now transferring out with all our accounts with them and we will reconsider our decision.
We did transfer some of our domains already to Namecheap and looking at their controlpanel it looks very nice, and their support is very fast. I received within one hour a reply from them which has not been marked as urgent.