Some crystals can't be put in water, or salt, due to degrading or scratching. Such info is available across the internet.
You can also use sound (voice, or whatever sound healing tool you have), breath, or hands on healing/intention (your skill set mileage may vary).
As you use them more and develop your psychic/energetic/auric sensitivity you will be able to feel when they need cleansing.
The full moon is not for cleaning per se, but charging, and some solar crystals (carnelian, tiger eye, gold, etc) won't respond to lunar energies as well. The dark moon is for cleansing. One can cleanse in the full moon, sure, a light bath so to speak, but the cleansing is secondary to the charging.
Personally I like dirt or running water.
If you're using smoke cleanse make sure that you are aware of the origins of the smoke you're using and ask yourself if you have permission from the spirits to use those items. There are big conversations about cultural misappropriation and theft around this topic that are worth exploring, and make your own informed decision about what feels ethical to you after researching what indigenous folks have to say about the topic. I'm not qualified to teach more on it, but only to point one in the direction of more learning and ethical practices. If you use smoke you've harvested yourself, this is a non issue.
Fire

such as a candle flame is also useful.