Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Durham to St. Augustine as you normally would.
In terms of precipitation, rainfall in St. Augustine ranges from 2.0 - 6.7 inches per month. September is typically the wettest month, when rainfall can reach 6.7 in. November is typically the driest time to visit St. Augustine when rainfall is around 2.0 in.
If weather is an important factor for your trip to St. Augustine, use this chart to help with planning. For those seeking warmer temperatures, July is the ideal time of year to visit, when temperatures reach an average of 80.6 F. Travelers looking to avoid the cold should look outside of January, when temperatures are typically at their lowest (around 57.2 F).