Confusion in changing name in N400

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Hello Everyone,

I Live in Southern California and I am going to apply for N400 within this week. I would like to change my name. My Last name is very long as I added my Husband's surname on my last name. My name goes like : Sally Ackerman Johnson Walter. Walter is my Husband's surname and Ackerman Johnson is my Maiden name. I want to change my Name to Sally Ackerman Walter. All my documents including current foreign passport and Greencard has my name as Sally Ackerman Johnson Walter as my name and it is very long and sometimes does not fit into credit cards, legal forms, Identity cards, etc

My question is:

1. I have heard that When I change name in California, California court will not issue name change certificate signed by the Judge. If that is the case, what can I submit as a proof for a name change ?

2. If I decide to change my name then I understand that I have to take Judiciary oath and Judge have to approve my name change first. Will I get Naturalization Certificate with new name on same day of oath or I have to wait for Naturalization Certificate in mail because USCIS may have printed old name in my Naturalization Certificate

Thank you for your response.