So, yesterday, I found a local bike shop: Rivendell Bicycle Works. Now it turns out that the majority of their business is online, but they still had two inner tubes for me and a patch kit.
I decided to keep one fresh tube at work and one at home, and then did some patch work on the existing tube. It's really a simple process if you follow all the instructions. Letting the glue dry a bit before application is key.
And it's all holding. The ride to BART yesterday was fine, and this morning has all gone well.