I working on my dual bachelors in political science in history. My university only offers courses on Russia after WWII, and when courses come about on Russia before WWII Russia-US relations aren't included in the course. When I attempt to literary works I don't have much success. I've heard only through word of mouth that Catherine the great had a hand in bringing the American war of Independence to an end by bringing the involved parties to peace talks. That Russia helped the Union during the Civil war. However, I don't know if these are true, but I would like to find out for myself. If any of you have any suggestions I would be very grateful.
Full disclosure, I am an undergrad majoring in history, poli sci, and IR, so my contribution is by no means professional. However, I do specialize in Soviet foreign policy and am taking a Russian imperial history course, so I have some knowledge on the topic.
I think that just typing "Russia and US relationship pre-1917" will not yield the results you're looking for. Instead, look for key diplomatic events between the US and Russia during your desired time frame. For instance, I searched "sale of Alaska 1867" in google scholar and found some sources you may be interested in:
NN Bolkhovitinov, RA Pierce, "Russian-American relations and the sale of Alaska, 1834-1867," (Alaska Limestone Press: 1996).
NE Saul, "Distant Friends: The United States and Russia, 1763-1867," (University Press of Kansas: 1991).
CA Manning, "Russian influence on early America," (Library Publishers: 1953).
I would try and get access to these books somehow and comb through the footnotes for further reading.
Hope this helped and good luck in your studies!