Well, the good news is that we don’t need to replace the whole, nearly 40 year old, heater. Not yet, at any rate.
The service technician said he believed that the moisture came from a rusted-out metal drip pan for the A/C unit (which we had on yesterday, so it had water in it). This is apparently fairly common on older units, and ours is by far the oldest he’s ever seen. In fact, he’s never seen one like it before.
Since there isn’t anything keeping the heater from working at this moment, he suggested we leave it at that and have someone come back to dig into what can be done about it tomorrow – at normal service call rates.