I can help you with this but if you would like you can also use an ftp client like firezilla ( or any other.) Also what is your website currently? Is it a forum? Because you can generate a back up to restore once you move to a different host if it is. Also if you have already changed your name servers to your domain you might have to use the ip that the domain has and the user name to view the website.
If my website domain was 123.45.678.90 and my account name was punisher I would be able to check my website by going to the link below
http://123.45.678.90/~punisher/
If my website domain was 123.45.678.90 and my account name was punisher I would be able to check my website by going to the link below
http://123.45.678.90/~punisher/