The mat initially is stick. You will see that it has been discussed on here a lot. And yes, you do need to use a brayer because just putting it down does not give good adhesion, even though it seems to be sticking.
When you remove any paper or card stock, use a little spatula, credit card, anything you can slide under there to release the paper. Even when my mat is not as sticky I still do that because I am in the habit of it!
It's just a good practice and you won't have the ripping and tearing that can happen with a new mat.
The other thing you can do is take a cloth (lint free) and blot it all over the mat and remove some of the sticky. But always use the brayer and something to release it and you will be fine and get much better cuts. Brayering is the key!
And on your other post, even though I DO have a Michaels or JoAnns, I still order online because of the choices and I can rarely find what I need. That being said, you are right. Most of us at least have the ability to make that choice and you are really stuck not having either near you. I think I might give up a crafting store nearby to live in Hawaii though! I still have not been there. It's on my bucket list!