NavList:

Your sights are an example of a “running fix”, such as when a boat takes a sight in the morning and crosses it with another sight in the afternoon to get the running fix. You have to determine the relative positions between when you took the sights using dead reckoning. 

You were a little vague about your position when you took the sights. I went to my iSailGPS app, looked on the NOAA chart and decided that 36deg 5.5’ N and 75deg 44.5 W would be a good guess for your first Assumed Position (AP). That exact position doesn’t matter much; what is important is the relative position between the first and second sight. Using the information on your moon sight you provided, I find:

Moon sight: Azimuth = 254.5 and Intercept =  0.57 miles. Pretty good sight!

I don’t know what method you use to work up your sights; obviously I didn’t use HO229. You can use a simple scientific calculator to work the formulas for the Nautical Triangle, but if you are not comfortable with that level of math, there are inexpensive apps for the iPhone that will work up these kinds of sights for you, and include all the necessary ephemeris.  If you insist on a paper table, the S Table is very compact and will let you use any Assumed Position. It is only good to one NM, but that should be generally good enough.  

Regarding your second sight, you said it was 2 nm due East. Looking at the chart, I think 1.72 miles is a better guess for the other side of Kitty Hawk. We need to find the Latitude and Longitude for this second Assumed Position, related to the first AP by your (car) dead reckoning. Obviously the Latitude is still 36deg 5.5’. A minute of Longitude is only a Nautical Mile (NM) at the equator. Elsewhere, as the meridian lines converge toward the poles, a minute of longitude is only the fraction, Cos(Latitude), of a NM.  So you have moved 1.72NM/Cos(36) or 2.1’ longitude, and the AP for the Jupiter sight is: 36deg 5.5’ N; 75deg 42.4’ W. 

With the moon sight Az and Intercept we could (but won’t) plot a Line of Position (LOP) through the intercept and perpendicular to the azimuth. You were somewhere on that LOP when you took the moon sight. As you move along a course 90deg and distance 1.72 NM, you want every point on that LOP to move the same way. Then you have a new LOP that you are still somewhere on. We could move the plotted LOP 1.72 NM east, maintaining the same orientation on the chart. But we won’t - too much work. Instead, we realize that the new LOP would be generated by the same Az 254.5 and Intercept 0.57 miles, but now taken from the new AP. 

With the Jupiter sight from the new AP I get: Azimuth 102.7;  Intercept 1.41 NM.

When getting a fix from two sights, it is best if the LOPs cross at as near 90 deg as possible. Your moon sight was pretty much to the west, and the Jupiter sight mostly to the east. The angle between them is only about 28 degrees. Those lines will give you a pretty good longitude, but any error in the intercepts will come out as about double an error in latitude. Now both the LOPs are defined by the Azmuths and intercepts from the same Jupiter AP. You could find the intersection of these LOPs by plotting them on a chart or plotting sheet. But I won’t do that either - again too much work. Instead, there are formulas in the Nautical Almanac for calculating the intersection of two LOPs from their anzmuths and intercepts. Using those formulas, I get:

Position: 36deg 01.6’ N; 75deg 41.7’ W

That puts us about 1.5 NM North West of the Wright Monument, if I didn’t make any serious mistakes in all this. It might be helpful, when doing practice sights, to write down the GPS coordinates to check your work.