I have suffered both conditions and can tell you that paleo by itself is not necessarily an automatic solution without a little further knowledge. Hashimoto's is an autoimmune disease and should be treated like all autpimmunitites which can take one of two different pathways and it is necessary to find out which pathway is causing your disturbance.
The guy that I am going to has a video on autoimmunity and he gives a brief explanation of the process here - http://www.askdrdanielroney.com/autoimmune-disease-and-treatment/
He will put one on a Hashimoto's challenge similar to the one described here - http://weblac.com/Hashimoto's%20Challenge%20Pack.html
... in which you take one pack for three days and the then the other for three days to see how you react.
Turns out I'm one of the rare cases that need both (twice as much of one as the other) but my energy levels have improved extensively. He is also treating me for the adrenal exhaustion using other means. But getting the Hashimoto's under control is the first basic step.
One of the first things he takes you off of is gluten which is an extremely important frst step and he is basically paleo, or maybe primal. The adrenal burnout is a slow thing to repair, but the a day or two on the Hashimoto's challenge can make dramatic change in the way you feel, usually by the next day.
So stick with paleo but that is only a first step. You may be able to find a functional medicine practitioner in your area. You could also google X-Viromen and X-FLM and look at the ingredients and see if you can simulate the challenge yourself.
