Experts and BHG readers answer.
Gardenia bushes that have grown to 5 feet tall will have extensive root systems. Typically the roots on a shrub like this extend out about 4 times the spread of the branches. That means that is will be impossible to dig up more than a small portion of the roots. You may have greater success in transplanting the shrubs if you do some root pruning this year, and move the shrubs next year. To root prune, dig down vertically all around each plant a foot or so away from the the main stems in all directions (effectively creating a soil ball about 2 feet in diameter). Severing the roots that extend out farther than this forces the roots to branch where the cut was made. That will concentrate the roots into a smaller area, and improve the transplant success rate next year after the roots regenerate.
The best time to transplant is when conditions are cool. In your area, fall is a good time to transplant woody plants. Late winter or early spring is also a good time.
Because you'd like to cut back the shrubs severely, the same time frames are the best to do that type of pruning.