Other than the nursing home itself and their headquarters, who do I contact about problems with nursing home care?

Start with connecting with an ombudsman. Then move to the regulating agency in your State.

In the paperwork you received from the NH upon admittance you should have the number of your State Ombudsman. You should be able to call the facility and get the number. They have to have it posted somewhere in the building by law. Can find the number in a phone book under State Government and online.