You seem to have adequate gas pressure (11" WC) cause the furnace runs ok, but when the furnace starts gulping LP, there isn't enough volume of flow to support anything else. I'm assuming both tanks perform the same way, but you should verify that. Assuming that, it sounds like (1) a partially clogged regulator, or (2) a crimp or clog in the supply line after the regulator. A little oil in the regulator can mess it up.
Have you run the water heater on LP? Does it have the same effect on the stove?