How can I get my staff to give me more support?
Your promotion is reality. You can expose your concerns about the situation and let them know that you need their help to make this work. It's better to get it out into the open than make believe that there is no tension.
Hold private meetings with each employee and let each one know how much you value his or her input and experience. Find out what is most important to each employee. Let each employee know that you are committed to finding solutions that benefit both of you. Show your employees loyalty, talk to them honestly, keep things on a professional level, clarify your expectations, and avoid nit-picking. Good luck in your new job.