Damp tinder is always a problem. You have to be able to both dry the tinder out and bring it to ignition. Which means a sustained flame.

The tampon you used is good for that. If you can enhance it by dipping the end in either candle wax or petroleum jelly.

That also works for tinder-quick. I personally carry a lighter and ranger-bands (strips of rubber cut from an innertube). They burn long and hot.