With aloes, crispy better than mushy. But give it time, it is healing already. Cold can do that to succulents, turning chilled red . Getting very wet and cold is really bad for them. If your temps are still cold, I would really hold back from watering. Just keep them in a warm bright spot, there is enough moisture in its leaves to get it through.
When temps are much more consistently warm, like in the mid 60's to 70's you can chop off that one with the leggy neck, about an inch or so from the actively growing green part, make sure to remove all dead leaves and stick in new soil. When you water it later on during the warmer months, make sure you allow the water to drain out, do not make the pot sit in water, so remember to empty that saucer under the pot.
