Yes, you can transfer the domain name from Intuit to GoDaddy, although I don’t know if I’d ever let godaddy build a website for me. Because… how to say lightly…. Godaddy sucks.
Anyway, hen you buy a domain, you have to have a ‘registrar’. You can host your website and domain at one place, and still have your domain registrar be elsewhere.
From what you say, it seems like Intuit is probably your registrar. You need to contact them and tell them you’d like to move the domain to Godaddy (unfortunately). You’ll then register the domain with godaddy and make them your new registrar.
This process usually requires your current (old) registrar to give you a code or verification credentials of some kind, to which will go to your new registrar. Intuit should be able to help you out, and sometimes its a headache.
Just know this – once you make GoDaddy your domain registrar, they don’t exactly make it easy for you to move it elsewhere again. Just a friendly heads up.
And regarding “deleting the old website”.... the website and domain are two different things. You can just download all your website files from the server so you still have the whole website in your possession, then just delete it entirely when you’re ready. You could then either upload the website files to the new web host when you move your domain, or start from scratch.