A full song played through a vocoder makes my throat begin to ache. I am not sure what I am doing wrong, but I do notice that I try to sing the melody instead of speaking it in an unusual way that is unlike normal singing, and this is hard to suppress. This unusual way feels very strained. Are there established vocoding techniques by artists who make extensive use of the tool?
I suspect that at the root of the problem is the fact that when singing into the vocoder, your brain doesn't get the expected feedback (the natural sound of your voice) and therefore instinctively tries to push harder on the voice.
For example, you know what happens when you wear enclosed headphones and someone talks to you -- you want to reply, but you don't hear yourself so well, and end up yelling at them...
And so I suspect that something similar is happening to you when you sing into the vocoder. Not hearing yourself causes your brain to instinctively force your voice to push harder or louder or strain itself more than naturally needed.
Now, in case of talking while wearing headphones, we usually quickly learn this new skill of talking normally even if we don't hear ourselves well.
And in your specific case, I think your problem may be solved by learning the same kind of skill, of using your voice in a relaxed way even if you don't hear it.
So, as a first step, try to always be aware of this phenomenon, be as mindful as possible, and sing or talk as relaxed as possible, until it becomes a habit.
Additionally, you may also try the following:
While using the vocoder, get a line from the mixer into your headphones, containing your clean voice. (For simplicity, use a second microphone next to the vocoder, just to send your clean voice back to you into the headphones). In this way, while using the vocoder you also hear your own clean voice in the headphones. This will help you become instantly aware of what is going on, notice any strain or forcing of any kind, and correct it.
Alternatively, use the second clean microphone line to record yourself (clean voice only), so you can listen to your clean voice later, figure out what you're doing wrong, and know how to do it better next time.
If indeed this is the nature of the problem, doing the above a few times should help you fix it permanently. And as an extra benefit, you may even discover some additional nuance in the way you talk or sing, which might further improve the overall quality of your music