It's actually the stark contrast between the Nile Valley and the surrounding deserts that made Egypt an ideal incubator for an early civilization.

Historians from Herodotus to Barry Kemp (drawing on his "Ancient Egypt: Anatomy of a Civilization" here) have noted the importance of the Nile. While Ice Age/paleolithic Egypt did have more of a savannah-like climate (per Steven Mithen's "After the Ice: A Global Human History, 20,000-5000 BC"), the shifting of monsoon patterns up to 5000 BCE caused much of the Sahara to dry out, increased rainfall in the Ethiopian highlands, and the transformation of the Nile from a series of interconnected marshlands into an annually flooded river. The combination of the drying landscape and the emergence of annually rejuvenated floodplains funneled the human population out of the growing deseets and into the fertile landscape along the riverbank. The people who came to live along the Nile found themselves in an enviable position: they had access to rich farmland that was re-fertilized by annual floods, and they were largely sheltered from raids and invasions by the natural protection afforded by the expanded deserts that enclosed the valley. This is reflected in the Semitic names for Egypt (Ugaritic: Mṣrm ; Hebrew: Mitzráyim ; Arabic: Miṣr/Maṣr), all of which mean "The Fortress(es)". Note that all this is before what we'd recognize as Egyptian civilization emerges.

Pre-Dynastic Egypt (ca. 5000-3000 BCE) is characterized by the emergence of local agricultural complexes with limited (though important) contacts with the Fertile Crescent. The Nile in this period serves another important factor: transportation. As the single major watercourse, it functions as a highway and encourages contact and unity among the emerging agricultural chiefdoms, with those that control chokepoints along the river gaining distinct advantages. It's ultimately the 3 phases of the Naqada Culture that spread through the valley from the Delta to the First Cataract (at modern-day Aswan) and give rise to the recognizable Egyptian culture of the Old Kingdom (ca. 2686-2181 BCE).

Throughout this formative period of the civilization, the Egyptian landscape looked much as it does today: the thin, green, agriculturally productive strip of the Nile Valley, giving way abruptly to the desert. There was probably a bit more rainfall to feed seasonal tributaries of the Nile, but these typically flowed quickly and briefly before drying out again (they exist today as wadis - dry riverbeds - still susceptible to flash flooding). All of this was reflected in how the ancient Egyptians themselves described their world. To them, their country was /kmt/ - often rendered as "Kemet", reconstructed as *kūmat (per A. Loprieno, "Ancient Egyptian: A Linguistic Introduction") - this name literally means "the Black (Land)", as in "the land of black (fertile) soil". Beyond "Kemet" was /dšrt/ - "Deshret", *dašrat - "the Red (Land)", i.e. the desert (and no, /dšrt/ and "desert" are not etymologically related, despite appearences). This informed the prominence of dualism in ancient Egyptian religion. "Kemet" was the realm of order and life, "Deshret" that of chaos and death. The regular flooding of the Nile and rejuvenation of the soil was associated with the myth of Osiris - a god associated with the afterlife, and with the introduction of agriculture and law to Egypt. The desert was associate with Seth - the brother of Osiris, the god of the desert, storms, and foreign lands. The abundance provided by the Nile was not only enough to feed Egypt itself; in later periods Egypt served as the "breadbasket" of the Roman Empire, for instance.

TL;DR - Yes, by the time "classical" Egyptian civilization emerged, the climate and landscape were nearly as dry as today, certainly dryer than the time of the "Green Sahara". Yes, the Nile was more than sufficient to feed an agricultural civilization. In fact, Egypt was the major grain producing region of the empires that followed after the pharaonic period.