Yes, boil, air dry, burn, boil, air dry
If you are careful you should be just fine as the coils are surprisingly robust. The main thing is to not pull on the wicks as this can drag the coil against the side of the head, damaging the coil, or causing it to short out.
I tried all sorts of methods before the dry burning because I wasn't sure. I read something about "never dry burn..." so avoided it, but now realise they were probably talking about cartomisers. If you try to dry burn a cartomiser it will melt and burn the foam material they are stuffed with - not a good idea.
I tried cleaning the coils as you descibe - does nothing.
I tried soaking in such things as alcohol, hydrogen peroxide, phosphoric acid, vinegar - does nothing
I tried scraping the coils with a scalpel blade - removes some stuff, but is risky, and doesn't remove enough to make a real difference.
Once I hit on this method it was a game changer. I have about 10 heads for 4 devices, and expect them to last many months with this method. (I'm also recoiling my heads now, but that is another story completely)
Since you've only got one spare coil and possibly no extra vaporisor device while you are dry burning you might want to accelerate the process. I really don't think the end result is as nice but try it and then you will know:
After boiling place the head on some absorbent kitchen towel. Pat the wicks down gently to start getting some moisture out of them. Blow any excess water out of the head. You can press down on the wicks to get more moisture out of them, but be carefull not to pull on the wicks. Inspect the coil and make sure it is in the right place.
When you start to dry burn it will take longer to get it to glow as you have to evaporate all the water out. Keep going with short bursts until there is no more smoke and steam, then make sure the coil is burned correctly.
After the second boil to clean it, follow the drying process above and install it in your tank, then fill with juice. The first few hits will be odd and tasteless - this is the water coming out of the wick. Flavour should return presently.
As I said, I don't feel the end result is as good as proper drying, especially before the burn, however it does get you most of the way there. Most people do not seem to be recommending the full drying time, or one or both of the boils even. I feel they are missing out. My senses of taste and smell are coming back now, and I'm getting quite particular about having the best vaping experience possible.