If I find myself caught in a sand storm, what should I do? Should I sit it out, or try to keep moving somehow? Can this depend on the severity of the sand storm?
Sandstorms (or, here in the southwestern part of the US, dust storms) impair the ability to see and breath. The winds also carry heavier debris. Because the storms often occur when it is hot, the wind can contribute to dehydration. The reverse is also true--cold, winter dust storms occur in some places, and hypothermia can become an issue.
Fortunately, most of the time, there is advance warning. Even if you don't hear the alert, you can see the wall of sand or dust moving your way.
You don't want to be moving around during the storm, the potential for reduced visibility is great. Even if you are wearing goggles, there's a chance you won't be able to see where you are going.
Bring your water canister and seek shelter--something sturdy. Even something that leaves you partially exposed will provide a degree of protection from sand/dust and debris.
If shelter is not available, then a higher spot is often better than a lower simply because there is usually less particle density at the top of the storm.
Move away from roads and freeways. In storms like this, drivers are advised to pull off the road, turn off their lights and keep their foot off the break. (Visibility can be so low that other drivers can't see the roadway. Terrible accidents occur because a driver mistook a car parked off the road with a car they thought was travelling on the road.)
Cover yourself--arms, legs, neck, eyes, ears, nose, mouth, etc.
Once you have yourself situated, then try to stay low to the ground and protect your head.
There are related conditions of which you'll want to be aware. These storms are often accompanied by lightening and rain. Particularly in desert regions, the rain can cause flash flooding.
You asked if the precautions can depend on the severity of the storm. Yes, the size and intensity of these storms vary greatly. When they are gathering steam, it's often not possible to distinguish the storm that has the potential to be more dangerous. Don't be misled, though--these storms are changeable and have the potential to become both fast moving and destructive.