Multiple force closes
After investigating further and looking at the partitions again, it turns out that I did have a totally full /data/data (as described in a previous question).
For posterity, here's what I did:
Because DiskUsage, the app I normally use to investigate partition usage, had locked up as well I fired up Terminal Emulator (luckily it was still working) and followed this advice I found in an XDA thread:

You can check [the partition utilization] by running this command as root:
df /data/data
If you see % used = 100 then this is your problem.

It was, in fact, 100% full. As soon as I moved some apps to my SD card my phone immediately started working again. I must've misread the free space of the partition when I first looked into it (doh!). The radio issues I was sure were involved ended up being a red herring.