Yes, although the connection isn't clear enough that Danes usually realise it.
Maybe because "it dries up" suggests something more extreme ie the difference between laundry dancing in the breeze and a landscape turning into desert. I don't know enough Danish to be sure, but I imagine that this difference could likewise be shown by adding a preposition? (Native speaker of English from Britain.)
Exactly what I was about to say. But I thought of it in regards to something spilled on the floor. Your clothes are 'drying', but the spill is 'drying up'. That's usually how the people around me distinguish the difference.