I recently decided to switch from 1and1 to GoDaddy, and in the process will be losing my domain name. I've planned on just buying hosting from GoDaddy and purchasing a name later. I just wanted to know if the website would be accessible without a registered domain name.

GoDaddy hosts thousands of sites. The server will not know how to forward requests for your website.

If I get a dedicated IP, would I be able to access that as if it were a domain name?

Only if you were hosting the web-site yourself. When you say GoDaddy is going to host your pages, I assume that you mean the web pages are going to be on their servers. They already have an IP address (actually ranges of addresses) which is unique to their network and their ISP.

A domain name is unique to a website. I can host as many websites as my hardware will handle, but the only way the server knows which pages to serve up is to have a domain name. Unless I only have one web-site per IP.

Why not transfer your domain name from 1and1 to goDaddy, it's complicated but doable,
here's a great tutorial seems to have just what you need, http://www.floppynet.co.uk/domsup.php