Why do I get dirty water for 10 seconds AND THEN THE WATER CLEARS UP?

Sounds like your pump is sucking up soil. You need a well man to check it out. If left alone this will clog up your faucets. It could be your sand point has developed a split that's letting in the dirt.
Is this a surface or a submersible pump. Let me know, Tom

What color is your dirty water, brown. Black or orange?