Are you still 'missing the MYISAM data files' for one of your databases? or is that remedied?
About backup for MySQL -
MySQL databases should be exported/imported. That's actaully how 'backing them up' works.
There are many tools; phpMyAdmin, MySQL Workbench (a free, cross-platform and very powerful tool for MySQL) the MySQL CLI (mysql.exe) or mysqldump.exe. Plus lots of free-to-expensive tools to automate all this. http://dev.mysql.com/doc/refman/5.1/en/ ... thods.html
Most of these tools produce .sql files as a default. These are text files with all the MySQL directives and data required to rebuild, recreate, or reinstantiate MySQL databases. The wonderful thing about this approach is you know you got a good backup, AND you can remediate after the fact.
Copying the datafiles is not particulalry risky if the two machines have absolutely identical setups and the MySQL servers are down when you do it. Any other scenario is asing for a disaster.