If your current host and new host both use cPanel, it's possible to make a full account backup and have your new host restore it. The other way is to have your new host do a direct cPanel to cPanel transfer which does not require making a backup and moving it over. Either way copies files, settings, logs and stats. That's the easiest way in my opinion. It's something I've done for several of my hosting customers.
The other way is to copy the folders and files related to AWStats from one server to the other via FTP. I'm not sure where all of the AWStats data is, but I know on my hosting servers there is a lot of data in the username/tmp/awstats directory for each user. I think that might be specific to cPanel servers, though.
Alternatively, you can do what lothos says. That can work. The only trouble is that logs can be huge depending on your site's traffic and the current server's settings. They can take a long time to transfer. Logs can also be too small if your current host rotates logs on a daily basis as many do. In that case this method would only show 1 day worth of info on the new server. When transferring via cPanel, you can actually delete the logs but still transfer all of the historical AWStats data.