> Were there books you could buy of these fairytales during that time in Russia?

Of course.
Very important difference of 1980 in USSR in comparison with today was serious lack of TV content.
There were 2-4 TV channels (depending on location), almost absence of stand-alone movie players and TV cartoons were rare and desired. Something about 10-20 minutes per day on state TV on regular days.
So, role of (more accessible) books in entartaining was more important and wide than now, especially for children.
For example library of my childhood includes classis russian fairytales, classic world fairytales (Hans Christian Andersen for example), more modern things like soviet translation of Wizard of Oz, The Wind in The Willows, many others and even first volume of Fellowships of the Rings. Even if you won't buy books you can freely go to the library and have as much books to read as you want for symbolic payment. One can't say that everybody read then, but more than now.
Oral tradition doesn't apply in my familiy to long fairytales, only short ones in very young ages. But parents read books of long fairitales often to children who can't read.