Paulownia is soft enough to mark with a thumbnail.
To use weight to help with ID, cut a rectangular piece appropriate for your best scale, measure the volume carefully, weigh, and compare with the numbers in tables or on the Wood Database. I have good gram scales so I can use small pieces but for more common household scales you might weigh the entire board. Density is usually given in oven-dry weight so even kiln dried wood will weigh a little more than the tables.
But again, looking at the end grain and comparing it with photos of candidate woods might easily rule out some species. Guessing the species by looking at the board is often just a wild guess since there is wide variation. Just look at the variation in hickory boards here: http://www.hobbithouseinc.com/person...cs/hickory.htm
Note that hickory often has dark heartwood, white sapwood.
Depending on how important it is to know this species and how quick you need it, you can get positive IDs by sending a small sample to the US Forest Products Laboratory. They will ID up to 5 pieces per year per citizen for no charge. It may take a few weeks to get results back.
JKJ