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'
it's passive voice though
You should say '제 남편은 땀을 많이 내고 있어요.' Still, '흘리고 있어요' is better.
I am a native speaker. 땀을 많이 내요 feels like he sweats intentionally (by working out, or sitting in 사우나, etc.). So, if there is no such context, 땀을 많이 흘려요 or 땀이 많이 나요 are used more commonly.
Why not " too much "?
많이 means a lot. 너무 means too much.
What is 'nayo' and why do we here use 'neyo'..?