The above is a 4 frame animation that shows my thumb striking the 12th string of a Universal. Notice that the pick is flat across the string and pushes straight through it. Don't scrape the string with the side of your pick.
The other fingers remain on the strings ready to play. If you are raising your hand then you need to slow your move down and let the fingers do the walking. Practice picking using a wrist move and one moving just the fingers. These are two separate techniques.
This shows the thumb picking a string and then moving to a new position. Notice that the thumb doesn't even block the string it picks. The thumb moves to pick the next string and the first finger comes down to block the string the thumb just picked. Also, with this technique my other fingers come back to the strings ready to play if I need them.
This is the techinque I use to pick a string with my finger. It is like scratching. Notice how it looks like a backhoe working. One of the reasons I hold my hand upright is to place the finger pick squarely over the string. When you have your hand turned you are only picking the string with the side of your pick. You should be able to do them all but the above will take you some time to work it out.
The finger does the work. The wrist doesn't even move. You can also use this techinque and the wrist in conjuction. I sometimes play licks that use a finger move and then a wrist move. A drummer showed me this techinque for songs like Johnny B. Goode that took a lot of stamina. He'd play a couple eighth notes with his arm and then some with the wrist. This kept him from cramping and slowing down.
The above shows the wrist getting into the action. The simple animation shows the picking of strings 11 & 12 then the hand changes positions to get ready to pick strings 11 & 10. The thumb and first finger are used on both.
BTW all of the above animations were created from the video: Secrets of the Wright Hand.