This is very strange. Dogs don't sweat like humans do, from the glands in their skin. They only sweat from areas which have no fur -such as their noses, and paw pads.
Dogs do have sweat glands in the skin covering their bodies but these produce pheromones, rather than fluids (like human sweat.)
There is a condition known as "hyperhydrosis" (swaeting too much) in dogs, but it is not from the glands under their body skin, but from paw pads.
That is as far as I know anyway. I don't know everything of course. But this sounds quite strange.
Are you absolutely sure this isn't very very dilute urine which is soaking the bed and couch?
If the urine was very dilute it may not have much smell at all.
Is your dog drinking a lot in the day?
Another possibility is drool. It's possible for a dog to produce large amounts of drool while sleeping.
I would say that if you are unsure of what's happening while your dog is on treatment for Lyme Disease, then the best thing is to get her seen by a vet -preferably the one who diagnosed Lyme and prescribed the treatment.
Poor doggy. Lyme is no fun. Is she on antibiotics for this?