The pills are neutralizing acid, but don't seem to be working because you have the start of Barrett's. A gastroenterologist specializes in digestion issues, which is what you have - an ENT specializes in ear, nose and throat - which you have a problem with secondary to the digestion problem. Starting with the ENT made sense because you didn't know what was causing the sore throat. But now that you know, you should stop seeing an ENT for a problem that requires a specialist - a gastroenterologist. Especially now that you have been diagnosed with Barrett's, which can be a precursor to esophageal cancer - this is not meant to scare you, only to inform you. Many people can keep the Barrett's under control without it ever getting worse. However, if you want to get a handle on your problem, switch to a physician who is better capable of resolving your problems and specializes in reflux problems - a gastroenterologist.
As to the way you feel on Nexium - here's what I personally experienced. I felt the same way as you when taking Nexium a few years ago, so I didn't take it for long. When I moved to a new town, I changed gastroenterologists. The new doctor wanted to put me on Nexium, but I explained I couldn't take it because of the way it made me feel. He said that some people can't take Nexium because they have a bacterial overgrowth in their small intestines. He ordered a breath test to determine if I had an overgrowth of bacteria. For the test, you drink a special drink, then breath out into a specially designed bag every 15 minutes for 3 hours. Each bag of air is later tested - if there is an overgrowth of bacteria, the gas it produces will be exhaled and captured in the bag (because the bacteria cause gas, that's why you feel bloated). The tests showed I had loads of bacteria. A week's course of a particular antibiotic cleared it up. Afterwards, I could not only take Nexium without bloating and discomfort, I could also enjoy dairy products for the first time in years. I wasn't really lactose intollerant like I thought, I just had too much bad bacteria in my system and dairy bothered it.
Remember that when you take PPI's, you neutralize the acid in your stomach. The purpose of acid in your stomach is to kill bacteria and prevent it from entering your intestines. Those of us on PPI's for long durations have allowed bacteria to enter our intestines that should have been killed in the stomach in the first place.
You may not feel heartburn because the PPI's neutralize the acid, however, you are still refluxing (that mean stomach contents are washing up into your mouth). Hence the nasty taste in your mouth. PPI's don't prevent stomach contents from going up your esophagus, they just neutralize acid.
You aren't crazy, you've just chosen the wrong doctor to handle your problem. I truly believe that everything you have been describing can be addressed by a gastroenterologist and a plan of action that will provide relief will be easier to work out than if you continue with your ENT.