Alright, it looks like Interserver has successfully copied The Ice Cave and my other subdomains to their servers. I've pointed my domain registrar's nameservers to the new service. Once everything has been moved over, I can cancel my plan with Hostgator.
I intend to put Interserver through the paces as I rebuild my fansite. Once I'm done with that and find everything satisfactory, I'll go ahead and move Damage Control Blog over as well.