I wear glasses, do you? The reason I ask is that they make it easier to deal with this issue.

Closing one eye and holding it shut is counterproductive, as you are using your eye/facial muscles to do so. An eye patch could help you, if you can wear one comfortably. I can't (long story), so what I do is fold up a tissue or two, lightly, and slide it inside my glasses so it gently holds the eye shut, there is just enough soft touch/light pressure to do the job. Sometimes I'll wrap part of it over the glasses on that side to help it stay in place, or just wrap all the way around that lens to block it out.

If you can comfortably wear an eye patch, but find you need the eye to be held closed, you can put a folded tissue or gauze or something inside the eye patch to fill the space and create the light pressure on the lid to let it rest.

Sometimes, just blocking the one lens, and letting the eye stay open, is helpful. Other times, it needs full rest and being closed. On a bad day, when I've over-used it to the point of it being twitchy, I'll lightly tape the tissue over it, so it's taped to my head instead of the glasses. I have found the light touch to be better than firmly taping and creating more pressure, just enough to let the lid rest closed.

If you wear glasses, before you get to the point of bad double vision, at first feeling of straining, you can also just stick a post-it note across the lens to block that eye while reading. You might need to block the whole lens, you might need to just block the outer side, the lower half, the inside, etc. You just have to try it and see what helps.

If you don't wear glasses, perhaps get a cheap pair of sunglasses and pop out the lens on the eye you want to keep using, and then block the other one with a post it or tissue/padding, etc and I've described.

Here's another trick my ophthalmologist has had me using for quite a while. When tired, overworked, etc, my worse eye wanders outward, and double vision results. He had me put tape, regular scotch tape, matte finish (foggy) kind, over the outer half of the lens. This way, when the eye wanders, it goes into the fogginess, and your brain then shuts off that signal and only sees what the good eye is seeing. Prevents the double vision, and lets the bad eye stop trying to stay in alignment, reducing the strain on its muscles.

You might need the tape in a different position, if your eye wanders inward (crossed) you might want the tape on the inner half of the lens. Or bottom, or top. But by taping only half a lens, you get to use clear vision when you have it, and are covered when the eye wanders, and the tape doesn't block out all light, so you aren't walking around with half of the world blacked out, and you still keep your peripheral vision around the edge of your glasses.

While reading, you should also experiment to find out the best position for what you are reading, where it is easiest on your eyes. It might be straight ahead. It might be down in your lap, or anywhere in between. It might be off to the side a bit. When your eyes get tired, where do they WANT to relax to? which way do they go? If you keep your reading material in their relaxed position, you may get to go longer before needing to rest.

I have been dealing with "wonky eyes" for quite a few years now, and my ophthalmologist, who is an eye muscle dysfunction specialist, has pulled out every trick he knows and then some, to help me keep going. Hope some of what I have shared here can help you. Double vision when you NEED to be seeing, just sucks.