Ok, so in spite of my misgivings, and the fact that NOTHING else is being offered or looks remotely like being offered, I went ahead and accepted the position described in my previous post.
My start date, depending on securing rental housing and their move-in date allowance, will be June 29. Somehow, I didn't expect that to be so soon. But it makes sense that they would want me to start quickly.
Wish me luck.
"The Director" (once again).