I finally got my wirewound resistors. I replaced the 1Ω resistor with a 12Ω and a 100μF in parallel. I replaced the 680Ω resistor and thermistor with a 1N4004. The Pignose works, and it gets plenty loud, but above a certain volume there is a fair amount of crossover distortion. I'm thinking the 12Ω is too large, so next I'll try something around 3Ω. 1Ω is too low with a 12V power supply. If you're using batteries, the 1Ω is probably best.
To answer your question, the goal of increasing the 1Ω resistor was to stabilize the bias. It's very stable, but the consequence is too much crossover distortion. I need to find some middle ground.