You really need to diagnose the problem so you make the best decisions about treatment, management, etc. The generic “liver problem” label isn’t helpful if your vet doesn’t have any idea what the underlying problem is or how to treat it, much less whether the lab values have anything to do with your dog’s current symptoms. What does your vet recommend now?
I’d insist on test for cushing’s disease, whether your vet(s) think he “looks” like a typical cushing’s dog or not. Incessant thirst is the hallmark of this disease and it warrants testing, IMHO, especially since the dog has repeatedly had normal urine glucose and makes diabetes less likely. Your vet can sedate the dog for the blood draw, if necessary. If your vet can’t or won’t help, then go somewhere else.
Constant, unquenchable thirst and the consequential endless urge/need to urinate is exhausting and a very painful condition for a dog. My old girl got this way when she developed Cushing’s disease and her life was dramatically improved with treatment…as did mine when I no longer had to clean up her endless–and unavoidable–accidents in the house.