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Peter, you asked:
"It seems to me that they actually measured the star's azimuth.  If that was 
indeed the case, does anyone know what they did with that piece of 
information?"

Each of those missiles had its own inertial guidance package. That has to be 
initialized with position and azimuth in order to get the missile from "here" 
--safe and sound in our submarine-- to the unfortunate place we'll call 
"there". If the azimuth is wrong by one minute of arc and "there" is a 
quarter of the way around the globe, you will miss the target by a nautical 
mile potentially giving the folks "there" the opportunity to return the 
favor. If you have time, an INS can "gyrocompass" to get an excellent 
azimuth, but that kind of time is not available for a missile launch. The 
submarine's own INS has a good azimuth, but in the early days of missile 
submarines, apparently it wasn't good enough to transfer to the missile. As 
it turns out, since the missile will be above most of the atmosphere about 
two minutes after launch, you can wait until then and use a single star 
tracker to get that azimuth information. I believe that's the solution that 
was adopted in the actual missiles with the radio sextant abandoned or 
relegated to a minor backup role.

As it happens, a couple of weeks ago while visiting the Submarine Force museum 
in Groton, I photographed the guidance package of the Polaris missile that 
they have on display there. I'm attaching it. The INS is in there somewhere!

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