According to Jesuit Luis Frois' history, when Jesuit Gnecchi-Soldo Organtino visited Nobunaga in 1580, Nobunaga had already seen a globe before and asked Organtino to bring it with him. At the meeting, which was conducted in front of not just Nobunaga himself but also a gathering of Oda retainers high and low (not mentioned exactly who), Nobunaga asked Organtino to point out on the globe the route he took from Europe to Japan and all the countries he passed through. With this, we can tell that Nobunaga had already seen a globe by 1580 at the latest and understood that the Earth is round. From that and existing artwork we know that knowledge of the Earth being round had already begun to spread by late Sengoku and early Edo.

However, in the founding years of the Edo period, due to the popularity of Neo-Confucian philosophy (specifically that of Edo Bakufu scholar Hayashi Razan), there were some controversies on whether the earth is a globe or whether the Chinese belief that the earth is a cube is the correct one. The knowledge that the earth is round stuck, at least among astronomers and cartographers. In 1670s, the Edo Bakufu's official astronomer and cartographer Shibukawa Shunkai convinced the Bakufu to reform the calendar due to the timezone messing up predictions with the Chinese calendar, and he made the first known Japanese-made globe in the 1690s, so we can say by the early-mid Edo period, the knowledge had gained official acceptance.

To the second part of your question, William Adams first landed and met Tokugawa Ieyasu in 1600. He continued to meet Ieyasu through the following two decades and showed Ieyasu globe and map of the world at these meetings. This was around the time Razan was rejecting the belief of the roundness of the world being spread by Jesuit converts (1606). So no, the Japanese did not learn of it through Adams. However he might have contributed to the knowledge's spread, and Ieyasu might have personally learned of it through Adams, but that we'll never be sure.