If there aren’t faculty with compatible research interests, why are you applying to that program? The match doesn’t have to be exact, but you will need to put together a committee who can mentor you through the dissertation process and evaluate the final product.
In terms of how specific - I think the more specific you can be, the better. I found it helpful to create a “funneled” statement that started broad and got specific.
BROAD: I’m interested in understanding how stigma or social disadvantage affects access to health care,
LESS BROAD: particularly in terms of how patients seek out health resources, how providers offer health resources, and how patients and providers communicate.
SPECIFIC: Specifically, I am interested in how these issues affect access to birth control and abortion care.
I also included a final paragraph about where I want my career to go where I talked about a couple of specific research questions or goals I would like to pursue. The committee wants to see a sense of direction because it says you’ve really thought about this.