If I build a campfire, but need to move on before it goes cold, how should I put it out? I wouldn't want to use my drinking water. Should I bring a covering to smother it with?
A large part of this falls down to planning as to where you build the fire in the first place. If you have got a natural water source nearby then build close to there which will give you an ample source of water to make sure the fire is doused. As an added measure, it also provides an easy source of water for putting out your fire if it starts to spread beyond where you intended.
If not, then at least try to build your fire on a stony area away from foliage where the fire may spread more easily, or will have more material to spread out of control if it re-ignites.
However, bear in mind these are just things that can help, they're not substitutes for properly ensuring a fire is dead before moving on. I'd feel comfortable smothering on stony ground and then waiting until things had cooled down, but at the end of the day if you haven't got the time to make sure a fire is properly put out and cooled down, you shouldn't have started one in the first place.
In short, you just shouldn't move on before your fire is out and has cooled down, but there are steps you can take for that process to happen faster.
If your fire has been burning for several hours in a stone ring, you need to deal not only with putting out flames and cooling embers, but with cooling the stones, which get very hot and could help an ember reignite. There's really no substitute for a lot of water. We use "washing up bowls" that hold about the same as a household bucket, and pour the water (taken from the lake) over the fire, stir, pour another over, stir, pour another etc until the inside of the ring is sort of ash-mud, then do one more over the stones around the edge to cool them. There's no way you would want to use drinking water in that volume. But when you're camping in a forest at the edge of a lake, that's what you do. (It's also what I do on my own front lawn in the stone circle I have there, except I can use a hose.)
Now if you made a small fire on damp ground, never really got to embers and didn't warm the stones, in theory smothering might be ok. (But not with dirt, grass, or leaves since those are ways to bank a fire and re-ignite it later, something you're presumably trying to avoid.) So might pulling it apart within a fire ring that is much larger than your fire. But why are you even doing that? To me having a fire is something you do when you're going to be at a site for a while (say, the evening) and putting it out is part of going to bed. Don't make a fire at your lunch stop, and don't make a fire if you don't have a source of copious water to put it out. As a canoer, I know that's easy for me to say, but I think it's the safest route. I've camped without fires during burn bans, including one year when we had two diapered babies with us, so I know a fire is purely a luxury. One we have to treat very seriously and that includes thoroughly putting it out.
A local source of water is obviously easiest - if you are near a stream - but if not, smothering can be really effective.
In planning, having your fire made on stones above damp ground will help, as you can scatter the embers a little without risking the fire spreading, and scattered embers will die down faster than a hot core of a fire.
Covering with damp soil can put out a fire quickly. Sometimes even really damp leaves or grass can be sufficient to put out a fire that is already dying down - but this will depend on your local environment.
In the absence of water, smothering with soil and thoroughly mixing the embers with the soil will effectively put out a camp fire.
A fire needs: fuel, heat, oxygen. By mixing in soil you remove two of those.
A fire is only out when you can run your hands through the soil and not feel any heat.