The first one with 'would' is fine. The second one sounds a little odd to me. Would here is describing a habit or a tendency. Might is more used for a possibility. I usually go out for a walk at night, but I might not tonight. I usually don't go out at night, but I might tonight.

For an English exam, the "I would" is incorrect. The first clause is in the present (I go - habit in the present) and the second clause is in the past (I would do - habit in the past). With more context, "I would" could be part of a correct type 2 conditional. "...but I would go out at night if my friends invited me. But sadly, they never do because they stay home with their families." In casual speech, some native speakers use "would" as a substitute for "might."

in my opinion, in the second one, the writer is less sure