The setup :
Domain : domain.com
Username : dom
make a temp directory on /home (i'll call it /home/cptemp/) where you can store the accounts files while copying.
Almost all space for an account is in their public_html directory, so we'll want to tar and gzip it and move that over using scp.
#tar czvf public_html_dom.tgz public_html
#mv public_html_dom.tgz /home/cptemp/
#mv public_html /home/cptemp/
Also often times large sites have large weblogs, which are packaged and transfered, so we take care of those as well :
#mv domain.com.gz /home/cptemp/
Now with these files in /home/cptemp/ , try moving the account over in WHM, hopefully it will succeed and set up the new account on the new server. Now we go back to /home/cptemp/ and scp the files to their new home,
#scp public_html_dom.tgz firstname.lastname@example.org:/home/dom/ (enter the root pass for the new server and let it copy the file over)
#scp domain.com.gz email@example.com:/usr/local/apache/domlogs/ (do the same as above)
Now go to the new server and unpack the two large files you just moved.
#tar xzvf public_html_dom.tgz
#gzip -d domain.com.gz
And that should do it.