I'm really quite surprised that you're having a hard time finding this stuff. A quick Google search for "Civil War soldiers' letters" turned up dozens of online collections hosted by the Virginia Military Institute. I would also recommend raiding the bibliographies of your secondary sources. Every quote had to come from somewhere. Find out where that is, and go there.

I would recommend the Virginia Heritage Site. You may be able to find some primary sources that fits your category.

Are you in a state that fought in the US Civil War? If so contact your university history library liaison, the public library, and the town historical society and ask them if they have any letters sent by soldiers in the civil war. This might not work if you are in a state that didn't exist in 1860.

Either way definitely at least contact a librarian that specializes in history at your university and they will provide you with a lot of ideas that you would have never thought of.

I just want to reiterate /u/TRB1783's advice to check bibliographies of secondary sources. Anything reputable and written by a trained historian will be extensively based on primary sources, and will point you to exactly where you need to look.

Check out the book 'All for the Union', it's very interesting and may help you a bit in your studies. It's a first hand account of a Union solider, Elisha Hunt Rhodes, and what he seen at Bull Run, Appomattox, etc. It's a book I picked up for my undergraduate degree, and I loved it.