You don't look very relaxed, which I understand if you are feeling like you will lose your sticks at any moment. We need to find a way for you to maintain your grip on the sticks without a lot of tension in your hands ... which is also evident by your elbows lifting high (out) when you play.
I would try this ...
Zildjian Drumstick Wax
I use this product, although it is under a different name (long story), and it works for me. My hands get dry and the stick wants to slip. The wax creates some traction on the drumstick and keeps it from slipping out of my dry hands. For me, even using lotion on my hands doesn't solve the problem because the lotion just softens the hands ... not giving any additional friction.
If the wax doesn't work, you may want to try gloves.
Some people sand the drumsticks to remove the lacquer ... which is aiding in making the stick slippery.
Work on relaxing your grip, especially around the fulcrum (index finger and thumb). You should only need enough pressure to keep the stick from sliding, but you can't really work on that until you resolve the dry skin ... or at least being confident that the stick isn't going to slip right out of your hand.
If you can, get with a professional instructor, even if it's only for one lesson ... and work on your grip. You'll be glad that you did and it will help you RELAX when you play. Right now, with the way you are playing, you are having to do all the work and not the stick or gravity.
