Also, is this a typo?
My final gravity when I bottled as 1.013. I took a gravity sample from the bottle and it was 1.023.
Because it makes very little sense.
What I meant here is my gravity reading before I added the beer to the bottling bucket, then when my beer tasted flat after 3 weeks in the bottle, I poured it out of the bottle into the test tube and took another gravity reading, just out of curiosity.
the confusion is that adding 5 oz of corn sugar to 5 gallons of beer only raises the gravity by about 2-3 points not 10. so either your sugar was not well mixed and the bottle you tested had way more sugar than others or something else was throwing off your reading. I don't know which. perhaps the hydrometer went out of calibration or was stuck to the side of the test tube a bit. or even just had some co2 bubbles clinging to it lifting it up out of the sample a bit.
try taking a bottle and putting it somewhere warm for a week. like 75f or even 80f. it won't really hurt the beer much but it should kick the yeast into higher gear. chill it down after a week or so and see if that bottle is carbonated.