I would like to hear some clarificatin as to what you mean by "droopy". If it's droopy as in wilting, then something is clearly wrong with the rose. Water if the it's dry, but if it's been well watered, then, you have a sick rose on your hand.
If, by droopy, you mean the rose is still healthy, but the canes are a bit slender and weak and whippy, then this is the normal state of affairs for a Teasing Georgia at this stage of growth. And this is true for a lot of shrub and climbing roses. It will need another 2-3 years to send thicker and stiffer canes. I would not prune too hard at this stage. The focus is to get the rose to have as much foliage for maximal photosynthesis in order to build up a good sized root system in it's first one to two years. It's only when it has a good enough root system that it will send up the more robust canes. Prune off any very weak, damaged or dead material. To prepare for the winter, shorten the remaining canes by a third of their lengths to reduce wind rock.