The "walls" are the foundation of your home and they rest on footings. Your footings will most likely only be just below the frost level.

You should NOT disturb the soil near the footings. When digging out a partial basement it is customary to leave approximately 2 feet of undisturbed soil adjacent to the foundation, pour a new footing there and a knee wall, then cap the space in between. You want an experienced contractor who will remove the soil and get the knee wall up promptly in order to minimize the potential for settling. No idea on cost, hopefully another poster has recent knowledge.

Many people then build storage cabinets atop the knee wall to disguise it. Hope this helps.

"When digging out a partial basement it is customary to leave approximately 2 feet of undisturbed soil adjacent to the foundation..."

Two feet is not a good guideline.

A 'bench' equal in width to the height above the new floor of the old footer is the usual minimum.

If the footer is 4 feet above the new floor height, a 4 foot wide bench is left.

You can go all the way to the footer if you are willing to shore up.

Around 4 foot sections usually works, but it depends on the type of soil and the loads.
You dig 4 feet wide up to and even under the old footer and poor a new footer.
Install support up to the bottom of the old footer, then repeat as required around the house.

It is not a fast process, and needs to be done carefully to avoid shifts in the old footer and collapsing.