Can you please explain what you mean by drawing with an appearance?
I tried the default stroke as per your suggestion - same problem.
See attached for the file I'm working on. It's a bunch of unstroked and unfilled paths. When I select a path, and add a stroke with the Control panel, it distorts.
Note the paths in my file are unclosed for a reason. They are motion paths for After Effects :)
2980_logostroke.ai.zip