log in to old server & go into the directory with all the content, domains etc (i.e. /var/www/html or /home/www) and type
tar cvf allmystuff.tar *
log in to new server & go into to the directory where all the content is supposed to be (i.e. /var/www/html or /home/www) and type
ftp ip-of-old-server
username
password
get allmystuff.tar
quit
tar xvf allmystuff.tar
If you bought the site, I assume the old domain won't be alive anymore, so there will be no duplication. If the old domain keeps alive and using the same content, then I guess you can't do anything