They look like clothes moths. Where in the house are you finding them?

Some of the remnants of those moths look like Indian Meal moths, one of the pantry moths, to me.

Closely resembles Plodia interpunctella, the Indian Meal Moth, this is widespread across the U.S. except for Rocky Mountain states and would certainly be common in New Jersey.

At an earlier stage, these would have been larva (caterpillars) in stored food. Some of your moths were looking for food to lay eggs in.