I've gotten paulownia here: Custom Millwork, Historical Reproductions, Natural Edge Slabs, Kiln Dried Hardwoods, 3",4", 5" Wide Flooring, Basswood for Carvers from Full Cycle Woodworks.
It's one of a very few places online that aren't just selling paulownia trees. Kiln dried and cut to order. BTW, like most woods, paulownia density can vary greatly. An earlier batch I got from another source was MUCH harder and more dense (33% heavier). That's not necessarily bad but you should discuss density with the seller to get what you need.
There are multiple sources for basswood and balsa. In buying balsa, I recommend you order "Hard Balsa". "Competition" or light balsa is meant for model airplanes and is very light and soft, requiring heavy reinforcement for crankbaits. I know - I've been working through a box of competition balsa for 2 years and am looking forward to it being GONE.