There are several issues to discuss in this situation.
If the probiotics, ginger tea, and tums were working, that would be fine. But if the heartburn is not well controlled, then it may be necessary to use a stronger medicine to suppress stomach acid. It certainly is true that any medicine can have adverse effects, but it is also true that sometimes certain medicines will work better than others.
There are also a couple of lifestyle changes that may be able to help. It may ease some of the reflux if you were to eat smaller, more frequent meals, and to avoid any food intake for several hours before sleeping. Placing 4-6 inch blocks under the feet at the head of the bed can also help limit the amount of reflux at night, which also may help with daytime symptoms.
The Aleve that is being used for the sciatica also can irritate any of the acid irritation syndromes, including acid reflux. There may not be much choice when a musculoskeletal problem is bad, but reducing the intake as much as possible and relying on medicines that do not irritate the stomach, such as acetaminophen/Tylenol, can help the heartburn.
From the standpoint of medicines to help counteract acid to help ease heartburn, it is also true that the Nexium and similar medicines are the strongest medicines that are available, and it may be that you can get control of the condition with a medicine that is less potent. The Tums is one of the least potent medicines to counteract acid. A liquid antacid may work better, partly because it neutralizes a greater amount of acid and partly because the liquid formulation coats the lower esophagus. The next type of medicines that are stronger than the antacids are the H2 blockers, such as ranitidine/Zantac or famotidine/Pepcid. These medicines were the primary medicines used for heartburn before the proton pump inhibitors were discovered, such as Prilosec and Nexium, and some doctors would still argue that the primary reason to use a proton pump inhibitor would be if an H2 blocker is not adequate. The recommended dose of ranitidine for acid reflux is 150 mg twice daily and for famotidine is 20 mg twice daily.
Is there any further information that would be helpful.