I don’t think it’s absolutely necessary that you read Burroughs’ first memoir before reading Dry, because Burroughs does a great job of filling in the blanks just enough to make the reader get what he’s referring to. I still highly recommend reading Running with Scissors first, anyway, to get the full story.
Dry takes place a few years after the end of Running With Scissors, and is primarily about Burrough’s stint in rehab, getting “dry”. He was working in advertising, and basically was given the choice of going to rehab because of his drinking, or losing his job. He “chose” rehab.