Is it possible to get a FUE hair transplant without shaving my head?

In your situation there is no need to shave your entire scalp. The only areas that must be clipped (not shaven) is the donor area on the sides and back, which from the photos already appear close to the length needed for extraction. The top of the scalp can be left alone as the grafts will be focused in the frontal area and hairline.

Thanks for your question.
If I understand you correctly, you are fine with shaving the donor region solely. Given your age, it makes more sense to focus on "almost" bald areas, and spare the sparse regions at this time round.
We can only shave the donor region and keep the hairs on top as it is now, and mostly work on the frontal part and mid-lock.
As your donor region photos are taken under bright light conditions, I don't believe it represents the whole picture, and therefore the donor looks thin/sparse.