I have to disagree with the previous posts. This sounds like overexposure or fogging to me if I am reading you correctly. You are getting an image out of the camera on the original paper that becomes largely black with only a few area remaining white when the sheet is developed Is this correct? If so, the paper is receiving too much exposure somehow.

You state you are using an "oatmeal tin" for the camera. Is this an actual metal cylinder or is it a standard cardboard oatmeal box? The latter needs to be painted black inside to become opaque. Even then, the lids usually leak light.

The description of your image with the patches of "white textured paper and nothing else" sounds like the texture that might result from exposing through the wrong side of the paper like when you make a contact sheet through the base rather than on the emulsion side of the sheet. Such an image looks very unsharp and mottled due to the paper fibers. Could that be what you are seeing?

By all means determine the pinhole diameter to try and narrow the correct exposure, but from what I'm reading, you have too much and not too little already. I suspect you are getting exposure fog rather than image exposure.