Start by providing documentation that shows both the old and new names. If you've recently obtained a passport, it may be helpful because it can show your old name as well as the new name as an AKA ("also known as"). If you are stonewalled, you may want to gently but forcefully give a rundown of your state laws that support your position. (You can research the law for your state at Nolo's Legal Research Center.)
If the person with whom you are dealing remains uncooperative, ask to speak to a supervisor. Or, if you have trouble at the local office of a government agency, contact the main office. You have the legal right to change your name, even if the people you're dealing with don't know your rights.
If you have still a hard time getting an institution to accept your new name, you'll need to go to court and obtain a judge's order establishing your new name. It costs a few dollars in filing fees and will take a little time, but it's something you can easily handle on your own. Once you have a court order, you probably won't have any problem getting your new name accepted.
When you're ready to change your name, see the Nolo eForm Declaration of Legal Name Change, which you can use to change your personal records, identity cards, and other documents.