I just kidnapped a friend's betta (she went out of town, gave us the keys to her house, and told me I could permanently kidnap him) that is having similar issues.
It's a culmination of months of bad water quality and being cold.
In a 1 gallon tank, you should be changing the water AT LEAST twice a week. Every other day is really what's recommended. Bettas thrive in temperatures of 76-82 degrees. There are heaters small enough for a 1 gallon bowl! I promise.
Start making 100% water changes every other day and you will notice a huge improvement right away.