I would leave the branches alone until next spring ....pruned now, the plant may or may not sprout new branches until spring
typically, adenium are in dormancy now - where growth nearly stops .. leaves yellow and fall off ....during dormancy watering is not necessary - and any water, can cause root rot, and will kill the plant
place in a south facing window (east or west facing windows are okay) for the duration of winter... when spring / summer arrives - place outside where plant will receive at least 6 hours of sunlight
never let adenium sit in water ...and never expose plant freezing conditions
when spring returns - prune limbs by about 1/2 ..... new stems - from 1 to 5 - will emerge .... this will result in many new stem tips and will give you more flowers ...
pinch out any new leaves that shade or crowd-out other stems ...as shaded stems may die out if shaded too long
your potting mix is going hold too much water anytime the plant is watered ....
I suggest you pull the plant from the pot .....knock off all potting mix .... place about 1" of rocks in bottom of pot .....then using *some* your old potting mix - add: sand, more rocks, lump charcoal, conifer bark - or - a bag of orchid mix
the new potting mix will not retain water and will drain quickly
@Stuart123 
with James wait until spring. And if that is your only one you can try to grow the cuttings so you have backups. 
