I registered my domain through speednames.com and now they won't allow me to specify the domain name servers of my web-hosting provider. They say that "they no longer provide this option." I can do webforwarding or point to the IP address of a dedicated server, but that's it. Has anyone else ever heard of this? It seems totally outrageous to me.

Not only is it outrageous its also illegal, assuming you are actually the owner of the name. If you are listed as the administrative contact you can swap it without even asking them.

I am listed as the administrative contact. Where do I go to swap it out? Since it wasn't registered through Network Solutions, Network Solutions can't let me make changes.

Who did you register the domain through?

use the whois/domain register under "Network Tools" on the left here...that will give you the info you need on who to contact as the Registra

Whois lists me as the registrant and the admin contact and lists hostmaster@speednames.com as the technical contact. The CORE Registrar is Core-11, or joker.com, but I don't have a password to access the domain at joker. So what can I do?

Try these two sites see if your situation meets their info... http://www.internic.net/faq.html http://www.icann.org
If you get too stuck I will transfer it out for you but you should be able to solve it