If you'll allow me to expand upon an answer I gave a little over 1.6 years ago here.

In short, most of them left.

Most of them weren't settlers as you think of them. Russian America, because it was operated by the Russian-American Company on a for-profit basis, needs to be compared to northern Canada and other territories operated by the Hudson's Bay Company — not the British colonies that became the United States.

The purpose of Russian America was to make money for the RAC's shareholders, not to provide settlement space for Russia. Imperial Russia, stretching from Poland to Canada, had more than enough space already.

Because of this profit-minded approach, there were never that many Russians in Alaska, and most "Russians" weren't ethnically Russian, even though they were identified as "Russian" on official censuses. Those censuses labeled Baltic Germans, Finns, Lithuanians and others as "Russians."

In any event, bringing more Russians to Alaska would have meant more expenses for the company. Instead, the company relied primarily on a network of creole Natives — half-white, half-Alaska Native. There were few — likely 1,000 or fewer — white Russians in Alaska at the time of the Alaska Purchase. Most of these, whether Baltic German, Finnish or ethnic Russian had specific ties to the Russian-American Company. When that company's assets were sold, they had no reason to stay in Alaska.

As Clarence Andrews notes in Story of Sitka, 168 people left on the Tsaritsa, which left New Archangel on Dec. 14, 1867. Another 300 people left the town on the Winged Arrow, which departed on Dec. 8, 1868. Other ships undoubtedly carried more people.

They fled despite Article III of the purchase treaty, which states:

"The inhabitants of the ceded territory, according to their choice, reserving their natural allegiance, may return to Russia within three years; but if they should prefer to remain in the ceded territory, they, with the exception of uncivilized native tribes, shall be admitted to the enjoyment of all the rights, advantages, and immunities of citizens of the United States, and shall be maintained and protected in the free enjoyment of their liberty, property, and religion."

Universal citizenship for Alaska Natives didn't occur the Indian Citizenship Act of 1924.

By the 1880 U.S. census, the first in Alaska, there were only 430 "white" individuals. Note that this does include a small Russian farming community at Ninilchik on the Kenai Peninsula. Ninilchik was a big exception, as Russian America derived most of its food from California and Oregon as far back as the start of the 19th century. The Russians were abysmal failures at turning Alaska into a self-sustaining operation.

Where did the Russians go? Many of them went to the Amur River basin, which had just been taken over by the Russians in a series of treaties with China. That area had much of the promise of Alaska, but it was on the same side of the Pacific as Moscow and thus seemed to offer as many advantages with fewer disadvantages.

Examine the case of Hampus Furuhjelm, the penultimate governor of Russian America. After he was replaced by a new governor in 1864, he returned to his native Finland for only one year before being promoted to rear admiral and appointed to governor the new Russian military province created in the Amur River basin. His headquarters was in the new town of Nikolaevsk na Namur, where he served until 1872.

Eugene Tidball's No Disgrace to My Country: The Life of John C. Tidball declares that by 1870, there were only 15 Russians left in Sitka, all suffering from disastrous poverty that denied them the funds for a ticket out. They were so poor that they were starving, and the U.S. Army had to keep them fed to prevent their deaths.

The Orthodox church was the sole lasting element of the Russian presence, though most of its officials returned to Russia after the purchase and were replaced by a new contingent of missionaries. Ironically, while the Russians had only limited success converting Alaska Natives during the period they owned Alaska, they enjoyed much greater success under American rule because American missionaries set such a bad example.