Cox stopped offering "personal webpage" with their residental service some time ago because of low demand. It wasn't a real webpage, just a place to dump picturesn and files. Now Cox Backup kind of feels that role.
Cox Webhosting with Cox Business is something entirely different. Its a professional web hosting product done with Network Solutions, a leading brand. Pricing depends on what you need, and while I "think" it supports ASP, you would want to call them and double check before taking any actions.
As for migrating, if you decide to go with Cox, let me know and I can give you tips on the migration process to minimize down time. But perhaps there is a easier answer. Do you feel comfortable telling what your current page is? Maybe if I see it I can think of a free option that will suit you better.