You need to cap your in-game framerate through the origin launcher...
Open Origin launcher
Right click on apex legends
Click Game Properties
Then open the advanced tab and type in the following command in the command line arguments
+fps_max 75
Then press apply and then re launch the game and check your FPS stability on OBS
If you are playing on a high framerate/high resolution monitor you will have to fine tune the settings based on your hardware
This is why it sucks using a single pc for streaming as you have to compromise your in-game performance in order to provide a proper viewing experience for your audience.
I have a 9900k, GTX 1080, 32 gb of ram and was able to find that I can run my game at 1440p at about 75 fps without any FPS skips or at about 100-110 fps on 1080p without FPS skips (this is when testeed on both x264 medium prest or nvenc new/old at 6,000 kbps bitrate)
If you play on 1080p you can maybe crank it as high as 120-130 fps on the frame rate cap within the origin launcher itself
This is one of the only games I've found that I have to cap my frames at in order to stream it at a descent quality. This tells me that the game has a lot of optimization to do as far as the hardware resource consumption side of things are concerned.
Hope this helps you man.