Ideally this is what you should be getting: https://www.youtube.com/watch?v=j_0QhZY-sEs
If it were poorly optimised then a similar result would happen.
What storage device are you using for your apex instilation?
Let check over some settings that may not be right in Nvidia control panel.
Right click your desktop and open Nvidia control panel; navigate to manage 3D settings the the program settings tab,
then in the drop box look for apex legends.
In the list of settings select and change them like so:
Power management settings = Prefer maximum performance
Shader cache = enable
Threaded optimisation = on
Another thing to check:
press the win key and type power; open the power and sleep settings then click additional power setting to the right.
next to the power plan you have selected click the change plan settings, then Change advanced power settings.
under hard disk expand the boxes and set the minutes to 0.
Then PCI Express and set the setting to off.
under processor power management set everything to 100% and active.
Apply the settings and see if there is any improvement.
while in the drop ship your FPS will differ compared to on the ground its just the way it is in BR.
Try this, just incase a rootkit is taking resources: https://www.malwarebytes.com/antirootkit/
Locate your games files and right click r5apex.exe, then open properties, click compatability and tick full screen optimisations and apply.
When you open apex could you open your task manager, click more details in the bottom left and expand the window a bit so we can see more of the list and post a screen cap here.