I went ahead and replaced the bulbs and set out upon a plan to replace the ladder. I looked at Home Depot today and discovered that I could get a relatively inexpensive ladder that was not actually rated to support my weight. It probably would...
I have now purchased an annoyingly expensive 2-step fiberglass ladder that is rated to hold me. The only thing more annoyingly expensive than this ladder would be the injury from a fall resulting from buying a ladder that came apart underneath me.