Yes and Yes

What you see is dust and some of it is on your sensor.

First make sure your lens is clean (glass - both ends).
While you have the lens off, make sure the mirror is clean.
You can likely clean the mirror with a squeeze bulb blower.
Do not use any type of brush on the sensor.

Your camera has a vibration device for shaking the sensor clean.
Look in your User Manual on how to use it.
Here is a review where they talk about the cleaning systelm: