Riding the Kontrol Freeks suggestion…I would start with a pair of the CQCx pads. They're mid-rise, convex, and grippy, giving you a much wider + uniform surface compared to the PS4 pads. These will be much more forgiving than the PS4 pads when it comes to fine motor control.
http://www.kontrolfreek.com/thumbsticks/fps-freek-cqcx.html
(EDIT: What I mean specifically is, because of the width and super grippy-ness, your thumb can be/can float almost anywhere on the pad, even at the edge, even with a tiny amount of force, and you will still maintain superb control!)
Definitely reduce your sensitivity. Drop down to 4 and play it for at least a week.
Lastly, you might want to look at SquidGrips. They are very grippy covers for the PS4 controller. You'll find you can relax your grip a bit using these, plus there's the added benefit of making the controller overall a bit larger in-hand.
http://www.amazon.com/SquidGrip-Controllers-controller-included-playstation-4/dp/B00KVKDNY4

Hopefully these and the other suggestions here help get you more comfortable in blasting faces off! :)

So what I did for my controllers was buy any of [these] (http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&amp;field-keywords=metal+ps4+thumbstick&amp;rh=i%3Aaps%2Ck%3Ametal+ps4+thumbstick)

After getting those on I got grips

Finally if you get sweaty hands while playing like a lot of people I HIGHLY recommend [squid grip] (http://www.amazon.com/SquidGrip-Controllers-controller-included-playstation-4/dp/B00KVKDNY4/ref=sr_1_1_twi_3?s=videogames&amp;ie=UTF8&amp;qid=1421194990&amp;sr=1-1&amp;keywords=squid+grip)

If you opt for the squid grip make sure you wipe your controller down with iso alcohol so theres no grease or crap on it when you apply the grips and read the damn instructions.

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Reddit mentions of SquidGrip for PS4 Controllers (controller not included)