Congrats for inheriting a Magento site :). I'm a big fan of Amasty's full page caching system. It works really well, usually without any tricky configuration.
On a single server installation, the cache files are often going to be in the /var/ directory.
If you're running two front end Web servers, then it is more likely that it's in a memcached instance or reddis location that can be easily accessed by both servers.
I would check the local.xml file in the /app/etc/ directory - this is usually where the location to the reddis or memcached instance is referenced.
Quick edit: I just noticed that you're running Magento Enterprise 1.13; that has it's own full page caching system. (Usually we wouldn't see Amasty full page cache running on an Enterprise site, as full page cache is a built-in feature of Magento Enterprise.) There's also another xml file to check in the /app/etc/ directory that deals with full page caching as well; I think it's the enterprise.xml file, so check there.
I hope this helps? - Jeff