Well, the good news is that you found your problem and the fix will be cheap! The thermostats work with wax that melts inside the sealed area causing th plunger to open and close. For some reason, you faulty thermostats appear to be sticking when opening and/or closing the way you describe the cycling. I'd pitch them both and try a new one from a different manufacturer.
For what it's worth, the AutoZone's private label Duralast thermostats come with a jiggle pin for both the 160 and 180 degree versions. I'd go with a jiggle pin version over one without any time.
Glad you found the problem,
Matt
