Sometimes it really is just body chemistry, though I completely understand not wanting to be on medication. You need some sodium in your blood so don't cut it out altogether because not enough sodium in the blood can kill you and no one wants that. 
My mom has edema that can't be explained and she takes a prescription diuretic; she has been tested for anything and everything that has edema as a symptom, taken off of medications that cause edema (she is Biploar Type 1), follows a low sodium diet, drinks plenty of water, and the edema did actually lessen in severity but it didn't go away completely and she does still need the medication.
Have you asked to see a nutritionist that has experience with conditions such as yours? Maybe they can help you find a diet that balances your body's nutritional needs with what your body is doing.
Have you been tested for any health conditions that can cause edema? Or are you on other medications that have that as a side effect? Work with your doctor to get off of those medications and on to something else that doesn't have that side effect.
Now for my disclaimer: I'm not a licensed healthcare professional. Do consult with one before you make any changes to your diet or medications to keep yourself safe and healthy. These are just things I've learned as my mom has been through a similar thing.
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