I feel your pain, I suffered a severe brain injury 3 years ago in a brutal car accident. Video games were part of my recovery process. Hand eye coordination took a long time to come around. To this day I still struggle to "see" things others just see.
My advice would be focus more on your enviroment look for cover whenever possible. Play the smaller legends stay away from Gibby and Caustic. This will make it easier in open combat to dodge behind a tree. Take your time, and make sure to do what you can to support your team. One thing I have found handy is keep an inventory of nades. Lob em and lob em again.
As far as hostility take it with a grain of salt, it is easy to be aggressive and hostile in any internet based enviroment.
One other piece of advice follow your teammates ues in game, if they are engaging everyone they see then follow suit. If they are pinging enemies but not engaging do the same if the situation allows. I played a ton of random games in season 2 and the one thing that struck me was how many players ran the spectrum on engaging or not.
As far as gunplay and getting used to it download titanfall 2 and play with the guns there in multiplayer or even game mode. These guns in Apex are just newer versions of those weapons.