So - what did I do to recover. Below I list everything I did and didn't do - I can't say for sure which element helped or whether it was a combination of elements - but I recovered.
1. I didn't delete any links, nor did I use the disawow file.
2. For all the links I controlled, I went in and changed the anchor text so that each link had a unique anchor. I didn't use anchors such as "click here". Instead I used actual search terms that people had used to find my site. So if my main keyword was "red widgets", the anchors now said "best red widgets", "buy red widgets", "where are red widgets made", "red widget" (singular), "best-selling red widgets", and so on. I was trying to get in all the variations and synonyms. Each anchor was really a partial match, and I used each anchor only once.
3. I made some more links, again with a completely unique anchor text
4. I built backlinks to the existing links to strengthen them
5. I increased the size of my website from 30 pages to 80+ pages, and took care to make sure the internal linking was robust. Again, when internally linking, I tried to use a wide variety of anchor texts.
That's it. That's what it took to escape from out of Penguin.