If you don't know whether or not you're going to have an exam next month, you use a 1st conditional: "If I don't have an examination next month, I'll go travelling." If you halve an exam next month, and you are imagining what you would do if this situation were different ( a hypothetical situation), you use a 2nd conditional: "If I didn't have an examination next month, I'd go travelling."

"If I didn't have examinations next month, I would go traveling." <- I think what the other dude has corrected is the right answer....

"If I didn't have examinations next month, I would go traveling."