It's acceptable but 흘려요 (spills) is more conventional way to say it this case. 땀을 흘리다 (more of a description / to be sweating) 땀이 난다 (to sweat). She is describing that her husband is sweating.
Because the equivalent to the english 'verb + ing' is 'verb + 고 있다' in korean. So the sentence would be something around the lines of '제 남편은 땀을 많이 난고 있어요.' (At least I hope that's correct..) If we use what HanBin said it would be '흘리고 있어요'
I think "My husband is sweating a lot." is the same as "My husband sweats a lot." Apprently, I am wrong in that thought. How?