Sounds exactly like the issue I have with American Truck Sim (DX9) and my R9-390.
Going AMD was a terrible mistake.
After installing Crimson 16.7.2 (since it gave me +4 FPSs on avg. in ROTTR), I have started testing the drivers with various options on and off.
I have successfully managed to run GTA V at 60 FPS avg. with Crimson 16.7.2 driver without stuttering! Here are the changes that I made in order to achieve that:
1. I have updated Intel Chipset INF drivers (but I believe that this had nothing to do with FPS increase).
2. I have set "Graphics Profile" in Crimson to "Optimize Performance" (this had a positive impact, but for little).
3. Then I went to Crimson > Preferences > Radeon Additional Settings > AMD Smart Now Technology > Disable Smart Sleep
4. Finally, the MOST IMPORTANT THING; I went to Crimson > Gaming > Global Settings > POWER EFFICIENCY: OFF! (this made me a miracle!).
After these changes GTA V ran without stuttering at solid 60 FPS with my custom/high settings without a problem (see table with results below).
It seems that the "Power efficiency" option, which was turned "on", nerfed my whole R9 380 card! This is too bad, since probably many other owners have similar problem and don't know for new extra options inside Crimson driver that are now turned on.
All in all, GTA V seems to be running pretty solid now. Slight frame rate drops occur while rendering explosions, which brings me to conclusion that the 16.7.2 driver is on average worse for about 1% - 2% than 15.11.1, but that is about 1-2 frames per second which can be tolerated. However, when I compare the minimum FPS, the 16.7.2 driver is 20% better than 15.11.1. It seems to me now that AMD has done good job with the new Crimson drivers. Too bad they by default enabled options that nerf their cards.
GTA V Benchmark results with Crimson 16.7.2 (and Power efficiency option "OFF")
min max avg 0 37 64,7 59,9 1 28,6 64 59,9 2 27,2 63,7 59,9 3 29,5 64,7 59,9