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Hedwig is the best variation. That's actually a family name of mine. It's my grandmother's middle name, and it was her grandmother's first name. I used to use it for myself in German class. It's very rare in Germany now; most of the people with this name are 80 or older. I think it's beautiful.
Edwige is also a nice variation.
I understand that you like the name Hedwidge. It's pretty cool, and you have a legitimate right to use it because it's a family name. However, because of peoples' ignorant naming preferences in the modern day, you may wish to stick with Hedwig.
I can also suggest to you the name Hathaway. If you are to etymologically trace the name Hathaway back a 1000 years to Old English, the name then, which had many variations, is very similar to Hedwig...one variation was Hadewig.