On both of the ones I got, the light did keep blinking with nothing attached, and I tried what you suggested as well.
Mainly I wanted to update: I emailed the vendor (I'm unsure whether I should mention them here) asking if this was a known issue and if they had any suggestions...it didn't even occur to me for a second that I was due replacements. Didn't even cross my mind! I just wondered if there was some fix they could suggest, and also figured they'd appreciate a heads-up if there was some ongoing issue with something they were selling.
However - I applaud their great customer service. I got an email back within a few hours saying they had very few issues with the batts they were selling and they put another two in the mail for me. So kudos to an ECF-verified vendor who stands behind their product! I will keep recommending them here.