You have to look for grants that support operating expenses, not programs. You are right, most granting organizations want to fund programs because it helps THEM tell stories of success. But there are a few organizations out there that recognize the dilemma you describe and are willing to help you over the hump if you have a promising mission and a realistic approach to growth and development.
Check out The Foundation Center. They have a wealth of information for people in your situation. You will find they are an invaluable resource no matter what stage your nonprofit is in. There are Foundation Centers all across the country; they offer training as well as information on most grants and grant makers in every sector. You will be blown away by all the resources—stay focused on your current challenge and don’t be afraid to ask for guidance. If there isn’t a center near you, they offer online help as well, and many of their resources are available in digital form.