Well, as Pharaoh might have done in a similar situation, I decree that my radio is as it should be and this power-off hissing was there all along. As for the R/F modulator installation, it works as advertised as well. Sound quality is ok, but there is no fading in and out of the signal or interference from FM stations like I had with the wireless model. But for $40, it is a very cheap alternative to any of the others that I've seen so far.

And many thanks for the VERY helpful radio removal instructions on the site and also for the help with diagnosing this issue (which is no longer an issue, so say I).