They do not need a trench, or certainly the drainage if needed can be like any house and any French drain if needed can be next to the foundation and not underneath it if that is what you are worried about.
A French drain under or next to the foundation is often done to help mitigate moisture wicking up thru the on grade floor. But remember you will also have a lot to think about on drainage so your basement is dry.
You could have a basement and still have cob or straw bale walls
Both need a good foundation and to be up from the ground to not get wet or wick moisture. As the saying goes, good boots (foundation) and a good hat ( roof).
I have not seen it done as cellars are very unusual in my area of California and also as it is more complicated than building without.
Just brainstorming in my mind, it seems having a basement would be easiest then by having a concrete foundation which you likely would be using for the basement anyways. Then you would have the normal way to attach the floor joists as you would now need to have a wooden floor or sub floor. You need to make sure if cob you have a way for the cob to"key in" to stay attached to your foundation of course so do research on what a concrete foundation without a basement is like for cob to make a sufficiently rough surface or other attachments keys. Lots of other details too like maybe the foundation is wider than the walls so the floor joists sit on it, then a lot of thought of detailing that area. Likely why I don't see it on the owner builds out here
out in the garden
I didn't do it. You can't prove it. Nobody saw me. The sheep are lying! This tiny ad is my witness!
Permaculture Technology Jamboree: June 29th-July 10th, 2020, Wheaton Labs