NavList:

Kermit, you asked " Between 17:00:00 UT and 18:30:00 UT, where do you start searching and narrowing your search to obtain the 2 solutions ? ".

I use Excel and compute the required values for a certain UT, and then my "zero-function" (corresponding to your f(UT)). Using the method of "interval halving" I got the following result:

UT=17h             f=2.7
       18h               -6.7  the zero crossing must be somewhere between these times
       17h30m         -6.9
       17h15m         -2.4
       17h7.5m        0.103 now we are close
       17h8m          -0.066
       17h7m45s      0.019
       17h7m50s     -0.0095
       17h7m48s      0.0017 and so on until
       17h7m48.31s  0.0000
       18h30m          3.3 as this value is positive and at 18h it was negative, the crossing is somewhere in between
       18h15m          -2.0
       18h23m           0.8
       18h20m          -0.28
       18h21m           0.08
       18h20m45s     -0.012 and so on until
       18h20m47.05s 0.0000

To determine the period of interest, I started to find out the UT of LAN, the time when GHA Sun equals 90° (which for a longitude of 90°W will give LHA=0). Then I found out that at that time the moon covered the sun's LL. And then I tested at some times before and after, don't remember now.

Moon's augmented semidiameter at exit time I got to 15.236', the same as you. I have not included contraction of semidiameters.
Lars