/app/tmp/cache/persistent
This directory contains a bunch of files, one of the files is:
cake_core_core_paths
This file contains a serialized array with information that is automatically "cached" in this file. Some of that information includes default paths. When I initially opened this file I could see some of my default paths were set to the old server. Once I deleted this file, then refreshed the page, a new file was created with the correct paths and everything worked as expected. So basically the problem was App::import could not load the correct file because it was using the wrong file path.