I am considering signing up with Hostony. I currently have a domain with another host and would like to start fresh with a new site on Hostony. Would I be able to sign up with Hostony without registering a domain, set up my new site and then later transfer my existing domain a couple of months later? Would that be the option to change DNS yourself in the sign up page?

yes is this option and you can do it
but... why "a couple of months later" ? You don't need to wait so much
dns changes of a domain is done in about ~ 48 hours... or you want TRANSFER the domain from your registrar to hostony ? (I don't know how much time takes this process...)

The reason I would wait is because my old contract with my old host is paid up until Feb. Might as well keep it active until it expires and it would give me time to build my new site and have it ready to go live.

Hello,
you can sign up for our hosting without restering the domain but you should provide us with your domain name as we can't set up a new account without domain name.
You can order 'modify dns' while ordering the hosting and we won't register such domain.

Best regards,
Slavik Koshelev