How can I keep my warm hands from making polymer clay too soft?
I might try two things, one handle the clay as little as required, the less you touch it the less heat it will absorb.  And maybe try something like Wristband coolers.  The link is to one brand, but the idea is that you can cool down your whole body with these.  It is a band that wraps your wrist (where you have a lot blood vessels).  I'll run my wrists under cool water on hot summer days and it will cool me off just as well as a bucket over the head, without the mess. So being able to cool down your hands should be possible and might work well enough to get you where you need.