Do plants use their stems to breathe for the whole plant, or do plants need to usually have slightly dry so they can grow better?
Generally speaking, plants do not use their stems to breathe but there are always exceptions (like cacti). The rule is that plants respire through their stomata, which are (typically) located on the undersides of leaves.
edit/note: A qualification should be made regarding my overly generalistic statement above. There are lenticels on stems which means that stems do breathe. But as a matter of comparison, there are (say on average) about 300 stomata per square millimeter of leaf surface, whereas there are (say on average) 1 - 5 lenticels per square inch on the bark (also, there's a lot more surface area on leaves). So stomata are by far, the primary portals for respiration (compared with stems).
Plant stems are typically better off dry but there are many exceptions, as various plants are adapted to various environments.
Roots do need air (roots absorb oxygen primarily through root hairs) because the plant (mitochondria) needs oxygen for cellular/overall metabolism. Plants also metabolize sugars and require oxygen to do this. So, you might ask yourself, why not take/use the oxygen waste from the leaves... or why is the oxygen that they produce from photosynthesis in the leaves wasted or not used for metabolic purposes. Again I'm sure there are many exceptions to the rule but the answer is because (primarily) segregating/sequestering hydrogen and releasing oxygen (waste) from the immediate vicinity of the photosynthetic reduction of water helps to bias the chemical equilibrium in favor of the reduction.
Cellular respiration (via mitochondria) occurs everywhere in the plant (not just the leaves), and generally speaking, the fuel is coming from the leaves while oxygen is coming in through the roots (edit/note: oxygen is also coming in through lenticles- located on stems, leaves, and roots). More specifically, sugars (primarily glucose), proteins and other organic molecules are coming from the leaves. Transporting glucose and oxygen together from the leaves, would probably result in premature oxidation of the glucose (in the phloem). In other words, to successfully transport fuels, you don't pre-mix them with oxygen.
For most plants:
The above-ground portions of plants should be dry the vast majority of the time. Fungus and insects love a moist plant. There is very little respiration/aspiration through green stems, and almost none through woody stems. Air flow requirements around a plant's stem are almost always pest-related.
Roots need oxygen. Aeration/drainage amendments are an important part of soil. There's no such thing as too much aeration/drainage, just too much aeration/drainage for a reasonable watering routine. Fabric/air pots have grown in popularity because of the slight (but demonstrable) boost in plant health they offer because of the increased air flow and the roots air pruning so the flow isn't lost. (Tree nurseries have been using similar methods for many years.)
If you've ever heard of something like "a rootbound plant can suffocate" - it's because their roots have grown (and died) so thick at the base of the container that airflow is prevented upwards and the plant dies. It's not a common thing - it requires a larger plant whose roots are demanding oxygen - but it is a thing.
There are exceptions to this, of course. On one end of the spectrum there are plants like Orchids, Succulents, and Mosses who drink water from the air and require oxygen-rich potting mediums and can suffer if their roots aren't allowed to dry between watering. Even conifers can suffer if their roots aren't allowed to dry out.
On the other end are plants that are either drought-sensitive or tolerate (or love) a constantly-moist growing medium, like Pussywillows. Even for these, aeration is necessary if watered with a drip line (air has to flow up while the water flows down).