I propose that the stamina consumption rates for walking and running are modified to produce a much higher difference. Some rough values could be:
running drain increased to ~ 150% of its current value.
walking drain decreased to ~ 30% of its current value.
running drain ~ 6 x walking drain.
It's some formula as far as I know, and nothing like "X for walking", "Y for running".
So assuming we have a variable "current speed" and a fix "running speed" value and a fix "walking speed" value, it could be (very roughly) something like:
drain = (((CurrentSpeed - WalkingSpeed) / (RunningSpeed - WalkingSpeed)) ) * (6 * walkingSpeedDrain) + (((CurrentSpeed - RunningSpeed) / (WalkingSpeed - RunningSpeed)) ) * walkingSpeedDrain
(I hope I have no flaw in this)
to accomplish "running speed drains 6 times walkingSpeedDrain compared to walkingSpeed".
However, that would be only a small part. You have to be sure that "CurrentSpeed" is an absolute value of course and the drain of sneaking is "undefined". (of course, it will get some value too with this formula)
The formula also doesn't take carrying weight or strength or anything else but the speed into consideration.
Actually I would even say that the "running stamina drop" formula shouldn't apply for walking. In fact, normal people can walk nearly endlessly.. the main reasons you are going to stop for are: your legs and feet will start to hurt after a while, but that isn't really the same as "getting out of breath".