Running setup in the new location might fix it (be sure you run it at /Applications/MAMP/htdocs/new-folder/setup/). If you have to create the setup directory, be sure it's from the exact same version of MODX.
If that doesn't work, you can manually edit the core/config/config.inc.php file -- check all the values there. Also check the config.core.php files in the root, manager, and connectors directories.
Delete all files in the core/cache directory after making any changes.
When it works, rename the original directories and make sure everything still works before deleting them.
discuss.answer
I must have been missing a line of code in the source, because I just couldn't get ModX to recognize the manager path. I ended up moving the install back to the htdocs directory and removing the references to /new-folder/ and now its working fine.
It's possible that deleting all files in the core/cache directory and clearing your browser cache and cookies would have fixed things.