I'm unclear why the question about how to sign your name was asked. I would assume you sign your name according to what your name is. If there is a reason not to do so, it may be prudent to consult with a professional who specializes in that aspect of self-representation.

Whether or not a correction is required for your name depends on what energies you want your name to represent. There are numerous reasons people have for changing their name, from professional to vanity.

There are misconceptions about what a name change will or won't do that are addressed at Misconceptions About Name Changes.

The primary thing to be aware of before changing your name is that the energy of the new name is highly likely to affect your life as long as you live. Name Change Effects talks about that. Be certain you want the new name before assuming it.