If you have acid reflux that gives you severe level of mouth ulcers in such a way that you are getting multiple clusters of canker sores on your tongue and you can't eat or sleep anymore, does that mean your sphincters are malfunctioning or are damaged and can only be fixed through surgery? Is surgery the only solution or can we do abdominal exercises that strengthen the sphincters; I have heard that you are also supposed to avoid any foods or drinks that trigger acidity but technically that is almost every food on the planet which is unavoidable and would probably result in deprivation of vital nutrients from your diet resulting in malnutrition and decrease of immunity increasing the problem even further.
Firstly, I would be reluctant to ascribe those symptoms (recurrent oral canker sores) to acid reflux. In very severe reflux acid can make its way up the oesophagus and can cause chemical burns to the throat and mouth, but there would also be other related symptoms, like nausea, indigestion, heartburn, persistent coughing (caused by acid irritating the larynx) and finally burns to the mouth and a taste of acid in the mouth, especially when lying down. Isolated persistent oral aphthous ulcers is not related to acid reflux.
Secondly, there are no oesophageal sphincters, in the usual sense of a muscular structure. The oesophagus is closed above the stomach when it passes through the muscular diaphragm, which applies external pressure on the oesophagus. This is called a 'functional sphincter', as opposed to an 'anatomical sphincter'. Thus, if this hole in the diaphragm is too big it won't be able to close the oesophagus fully and there will be some reflux of stomach acid into the oesophagus. Depending on how large the hole some of the stomach might be able to go through it, what is called a hiatus hernia. There are no exercises that will close this hole in the diaphragm.
Thirdly, if there is a strong history of typical symptoms then a diagnosis of reflux is fairly straightforward. However some tests are usually required to determine severity and possible treatment. Surgery, in severe cases, is recommended and has good results. There are medications that will help lower acidity levels and can eliminate symptoms in mild cases. This is best decided in consultation with a surgeon. A trial of medications is often worthwhile, to assess response to treatment, before surgery is considered.
Fourthly, diet plays an important role in the management of reflux. Anything you chew or swallow will cause your stomach to produce acid, but there are certain foods that cause much more acid to be produced or are actually acidic themselves. There are many excellent resources on diet available online or from your doctor.