I'm so sorry this is happening.
The thing about tension headaches is that the cause is not really well understood, and treatments are few. The thing that's always worked best for me is excedrin extra strength, which is aspirin, tylenol and caffeine.
Massage can also help, or anything that relieves stress and helps my muscles relax naturally. A scalp massage is the best thing on earth.
The antidepressant can help, depending on what it is, though I understand totally if you don't want to take it. Gabapentin is so sedating, and you may get used to it and have less drowsiness, but it's almost impossible for most of us to get to that point since it makes us non-functioning for so long.
Unfortunately, it seems that once you start getting headaches, it's so hard to get them to stop.
One thing to do is pay attention to what you are eating and drinking. Make sure to stay hydrated.
Things I do that may or may not work for you:
I use Bio Freeze spray on the back of my neck when I get one. My sister uses Icy Hot spray. They are similar. It helps relax the muscles.
Sometimes heat on the back of my neck helps, sometimes an ice pack does.
A hot shower, letting the warm water just hit you on the neck, shoulders and head.
One thing to check is your pillow. Make sure it's supportive enough, and doesn't need to be changed out. I have allergies to everything, and my headaches get worse when my pillow starts collecting dust mites. I get a new pillow every 6 months or so, but you may not need to do that. I also use allergy covers on my pillows. You can get those anywhere (Walmart, Target, etc.) for about $10. You also don't need an expensive pillow - just a hypoallergenic one that's good for the position you sleep in.
Feel free to get a second opinion from another neuro, too. Just because one says one thing doesn't mean that's the only option.
Hang in there. Headaches suck.