I'll give [edited] Hewlett full marks for his never-say-die attitude re teaching in China. Glad to see he's almost back here.

It appears that his recent woes were caused by a dishonest Beijing recruiter who strung him along with false promises of a full-time teaching job. I was in a similar situation a few years back. My problems at that time were due to a Chengdu recruiter screwing me over by not providing so-called guaranteed work that was promised me.
I also teach on F visas, which makes it more difficult, though not impossible, to find decent work. China'a a great country for teachers who are not properly qualified in terms of academic qualifications - we can get white-collar jobs that pay reasonably well as foreign teachers here in contrast to lousy low-paid unskilled positions back home in the West.
My advice to teachers without uni degrees is to contact smalltime agents based in provincial capitals. They tend to be relatively honest and charge their clients lower commission fees. I use three such agents whenever I'm looking for a new job. So far, so good. Avoid bigtime agents, to them you're just a number, and they'll say anything to keep you on their books!