I'm going to go ahead and replace the igniter, since it's only a $30 part and fairly easy to get to.
I will hold off on replacing the control board for now, but it's going to be my next step. We just replaced our old converter and the new one is just So much better, it's really shown me what 30 years of advancement in electrical engineering can do.
I know that it shouldn't keep blowing without ignition and that it should have a set number of ignition attempts, and it's clearly not doing that, which shows me there's something wrong with the brain. You all are pretty convincing about replacing it.
We're looking to get our first snow next week, and while our electric space heater will keep us from freezing, I think a furnace is going to be Really nice to have in the North Idaho winter.