Installed Acymailing 6.5.2, the JCE plugin for Acy is present (could be from previous installation of 5.10?).
JCE editor loads the template fine but there are no options to insert Acy tags anywhere. I think this is what the OP is stating. The editor is loading the HTML code in and the JCE editor is functional, its just not useful because none of the Acymailing functions are available.
Whereas if you create a blank/new template, you get the options to insert blocks, template tags, etc.