Did Privates in the Union Army have rain coats or something to keep them dry when they were in the rain? I don't mean when they were sleep but when they were marching, etc.
No, they were issued oil clothes/gum blankets. They had many uses. They had grommets on them. They could be wrapped around you like a poncho, used as a ground clothe, a makeshift shelter, amongst many other uses. They were either oil painted or rubber covered canvas. Here is a photo of my reproduction Federal issue gum blanket, this is a painted version.
These were also prized by Confederate soldiers when they could get their hands on them, they weren't typically issued these. When they could acquire one on the field they held onto it.
Stonewall Jackson was wearing a raincoat the night he was fired on by his own men at Chancellorsville. The picture of his coat I saw suggests that it was probably rubberized much like the poncho in the previous picture.