But also I wonder if the CIA keeps tabs on Carrie while she's in Russia. But she does seem to live a normal life with Yevgeny so as to appear that she has truly moved on and turned her life around to live in Russia.

Hmm, I may need to re-watch because I remember Carrie meeting up with an (American?) asset in a washroom so perhaps the CIA knows Carrie is a spy but they're not disclosing that so as to put up appearances in Russia?