I think 寒いですから or 寒いので would be better for consistency with the ending verb 着ています, and it totally should be accepted if you wrote that. Maybe Duolingo just wants to show us that です isn't necessary with i-adjectives. 寒いから上着を着ているんだ。
See ASleepingRock's comment above. さむい ですから is perfectly acceptable. だ is just the informal version of です so I think it should be acceptable as well. Remember, です is a being verb usually translated as "is".