We can rebuild it
I gave my old bike a more thorough examination today, and things are better than I initially thought. The rust I saw on the tire rims is confined to only one spot. Overall, it was mostly dirty and covered with spider webs. I cleaned off the majority of these and looked over the major mechanical systems. The brakes work, the drivetrain is in good shape. It looks like I can get it into ridable condition for the cost of a couple of inner tubes... about ten bucks. I'm going to give that a shot — if that doesn't do it, at worst, I'll have a set of inner tubes I can use later. Plus, I'm definitely more of a 'use it up, wear it out' kind of person than a 'throw it away and buy another one' kind of person.