I'm on a PC with Oculus, so I'm probably not even allowed to post here, but here goes:
I experience something similar as the OP. I also see horrible FPS from time to time, with double vision all over. It typically happens when using the Galaxy Map, after returning to the cockpit. When that happens, in many cases, I can get perfectly smooth performance by quickly switching to System Map and back. Looking at CPU and GPU performance in Task Manager doesn't show anything alarming. Also changing graphics settings to high/low changes nothing to the better. It seems that it has to do with memory swap. I don't know for sure but it could be processes in the background like loading the next system, or in the case of Galaxy Map, loading it and then loading the cockpit after plotting a route etc. I remember from the PS3 days, how packed everything had to be on the PS. Textures were packed to the last pixel on some of the later PS3 games, and since PS4 (I had one and liked it) is getting old, that might be the case again.
Try the System Map quick toggle before jumping, and see if it changes anything. In that case consider reporting it as a bug.