Pretty sure the solenoid is dirty and stuck. Some have been able to take it apart and clean it. My boy tried, put it back together and it jammed again in a day or two. He said it wasn't worth it to him, he bought me a new one.
He also installed an inline oil filter in the return line to the transmission. No more bad solenoids now.
My truck would start out in 3rd gear. As you said, if you shifted it to low, you could get it to go in 1st or 2nd, I don't remember which now.