i just went through the same situation with moving from justhost to hostgator. I had 4 blogs and 1 wesbsite. one domain had domain privacy and it was a pain in the butt to get justhost to remove it.
after many phone calls and emails, justhost finally removed the privacy and provided me with an EPP code, which i provided to hostgator.
if your domain has domain privacy, then that could be part of the problem.
hostgator transferred 1 website with alot of files and 4 blogs. everything worked fine, except for the blog that had the domain privacy.
once that was taken care of, everything worked fine.
i purchased a wordpres plugin called backup creator.
it allows you to backup, restore or clone a blog and it works great.
ask your new host if they can find out if you have domain privacy enabled on the old domain. it is easy for them to do it. it is some whois link.