Based on your description of Amy's signs (primarily the leaking urine while asleep and dribbling urine in the house), I agree that it sounds most like urinary incontinence. There are many potential causes of urinary incontinence in the dog.
It sounds like your veterinarian has screened for one common cause, urinary tract infection. While it's great news that a urinalysis was recently negative for infection, I will caution you that occasionally UTIs can be missed on routine urinalysis. The most sensitive way to detect UTI would be to do a urine culture, which is a test where they try to grow any bacteria that may be present in the urine (takes 5 days).
Another potential cause, particularly if Amy has had this problem since she was a puppy, is something called ectopic ureters. The ureters are the tubes that connect the kidney to the urinary bladder. If these tubes develop wrong such that they insert anywhere other than where they are supposed to, we can see chronic urinary incontinence. Ectopic ureters can sometimes be seen on ultrasound or on x-rays with special contrast agents injected.
A third cause of urinary incontinence is neurologic disease. Your veterinarian can do a thorough exam on Amy and make sure she isn't showing any signs of neurologic dysfunction.
Lastly, once other causes are ruled out, we are left with Urethral Sphincter Mechanism Incontinence (USMI). This is a condition that is occasionally seen after we spay dogs (so if Amy's problems developed after her spay, this becomes more likely). The medication your veterinarian recommended is intended to increase muscle tone in the sphincter that controls the urethra. These medications usually work very well, however they do need to be given lifelong to maintain control.
In conclusion, there are a few other conditions that Amy can be evaluated for prior to starting this medication. However, if costs are a concern, there is little harm in starting the medication on a short term basis to see if it helps her.