In reply to lnn:
I agree with Inn. It is true if you use walking poles your hands tend to get colder than otherwise in my experience.
I can think of a few reasons.
First, because your hands are positioned higher than just hanging freely, the circulation is limited more than otherwise, hence the hands tend to get colder.
Second, if you use the wrist loops, that will be another hinderance of the circulation, hence the hands get colder.
These two reasons are exactly the same as in climbing with leashed-tools and leash-less. The hands stay much warmer in the latter, because with leashless, (1) you don't have a circulation-constricting wrist loop and (2) you can shake out your hands easily, so you do so a lot more frequently.
Third, if you don't use poles, you can stow away your hands out of wind (to some extent) in the lee side of your body. Also, you can curl your hands inside your gloves to warm up easily. If using poles, those things are not practical, hence your hands (in gloves) are inevitably exposed to the wind and you can do very little to get the circulation back.
Fourth, although the handles of most walking-poles are reasonably insulated, they can not be better than holding nothing, namely holding your own hand (by gripping).
Fifth, when people use a pole(s), there is a tendency they don't grip but keep the hands loosely open, particularly if they use the wrist loop (strap). That does not do any good to keep the hands away from getting cold. Better keep the fist, because by doing so fingers are touching each other more or less, which helps keep the warmth.
Personally I use poles regularly, if not always, and I have got very cold hands. So, what I do is to keep the above-mentioned points in mind, and try not to do so.
For example, use a shorter length of poles (anyway, if your elbow is acuter than 90 degree, I don't think it is an effective way to use the poles). Not use poles on easy terrain and stow my hands away from the wind, such as, back of my hips or in front of my waist, depending on the wind direction.
Or, simply to walk faster can do a trick in some circumstances, as it enhances the circulation.
There are loads of tricks, which I have experimented in desperation and found over years... I am sure it is the same for any experienced winter activists. To stay warm and keep your extremities away from being frozen (it is impossible for me to keep them warm... Best I could do is to prevent them from getting useless) are a skilful job, at least for some people like me.
I hope you will find a way!
Masa