Well, Simple Simon said to me, "a little knowledge is dangerous", and the only thing I can say is, "a mind is a terrible thing to waste" so here's 2 of my senses.
As others have already said -- Enter the zip code for a Dish 500 and you will get the skew, elevation, and the azimuth as magnetic (and maybe it's updated with the annual magnetic deviation).
90 and 0 degrees skew: The Dish 500 skew set to 90 technically = 0 degrees, i.e., neither a + nor a - offset.
The Dish receiver calculations will actually give you the azimuth for your location for a single bird at 114.5W (half way between 110 and 119). Likewise, the skew is the happy middle between the 2 birds as calculated from various programs. But setting the skew to 90 (0 degrees) when using a single LNBF on a I-Bracket to receive a single satellite should give you an excellent signal after adjusting the elevation and azimuth. For such a single satellite dish, you could change the skew as provided by various programs, but will have to adjust the azimuth and elevation. This could be helpful with a limited view and not much room to relocate a dish, but moving the family or changing program source may be better alternatives.
So to answer your question,
"When I point the DishPro 500 to the suggested azimuth and then adjust the skew (by an off-axis rotation of the dish, not the LNBF) does this change the actual "looking angle" as it appears to do?
Yes it does.
So in response to your comment,
"I believe that the skew adjustment changes the looking azimuth by an unknown (to me) amount and thus need to determine what the actual change in looking azimuth is for a given amount of skew change if my assumptions about the geometry are correct."
Although I feel your assumptions are correct, I don't believe doing so is your solution. I believe if you make any significant changes to the skew, and then compensate/correct the angles by adjusting the azimuth and/or elevation, you will effectively be changing the arch for the locations you are looking at, instead of looking at the normal arch of the satellite (Clark's) belt. In other words, if you get 1 bird, you will be looking for the 2nd bird either below or above the belt.
Although it sounds like you are going to use the correct settings and look for the best opening, maybe the above will help if you decide to experiment.
Good luck.
NOTE: I never took geometry and slept through the Louisiana math classes while in high school (but got a 100 on every weekly test).
Multiple wrongs often equal a right.