I think some exhaustion is normal, especially since you have been increasing your LR schedule and level of training over a period of several months. Although you are feeling tired, just have confidence that you are better prepared for the half.
In terms of the taper period for this final week, there are many different opinions and it really is a more individualized choice. Some find it more beneficial to go back to the early or mid-sections of the training schedule and then only scale back a few days out, while others take additional rest days and a few even keep a regular routine (slightly reducing a few miles) right up to the event, so your comfort level is what matters the most. Certainly add some runs if that will help your energy and motivation levels, but (in the alternative), you will not lose the endurance you have been building if your body needs some extra time off before the event.
I am one of those runners favoring a limited taper for the half, but that is mostly because I actually feel more tired and nervous if my running schedule is disrupted too much. Again, it is highly personal and you will be fine whatever your choices in these final days.
Just go and enjoy your first half...start out slow, find your pace and have a wonderful time. Then celebrate your accomplishment! It is a big deal and you should be proud of all your efforts.