Shaina, it may be difficult for a northern gardener to suggest what might be best for you in such a warm spot....and sure, i understand that parts of the State might get down in temperature at night....but that can be to the geranium's delight, plants do like cooler temperatures at night.

Dropping branches. I dunno..
But you should appreciate that plants at this of season are slowly going into a dormant state...they require much less water than they used to...less light is given from the sun because it too is going down.....and its intensity is going down every day and until about mid February does it begin to come back up north.

So during the winter, do give your plant as much light as you can....the western exposure is good.
But even that, you should expect your plant will maybe begin to wane, might get straggly due to the loss of sunlight.

I think, since you cant possibly put your plant into a place where the temperatures are below 40 F...to keep until spring, your Arizona plant has to be kept going.
I think, and I'm only thinking out loud, in Arizona all you can do is try to keep your plant going as best you can until the sun does return some intensity.

You can, to make it work less, cut it back...you might at this time, check on the roots..make sure they are not in any way rotting from maybe being given too much water.
Cutting back according to the amount of roots can help.
This will encourage the plant to produce new growth when the sun does return. Until new growth is in evidence, don't fertilize your plant...that encourages new growth that will suffer due to the low sun.

Watch your watering. Allow your plant to dry down between waterings and when you water, water to drainage....and dump the excess. Don't leave your plant sitting in its drainage water.

If...by chance, you are on a well...and possibly your water is treated by a water-softening agent (maybe salt), don't use such water for your plants. Go off the system, find a tap --maybe outside...and use that water. Let your water gain room temperature....again, I'm thinking the water is cold...in Arizona...but plants prefer water at room temperature.

The other way is to take cuttings from your mother plant.
But, any cuttings should be taken from a plant in the best of health...otherwise the babies will be in poor health.