Translation:Read both books before the next class.
It's partly a question of idiom and partly one of specific vs. general. "The next class" is very specific - it's not stated, but classes are schedule, so one would know exactly when the next class is to start and how long it will last. "Next week/month/year/century/etc." are vague and general - we may know when it will start and how long it will last, but we don't know whether something would be due precisely by the time the week/month/etc. starts, or some time during that week/month/etc. For example, in a business setting, getting something done before next week may mean before Monday (the 2nd day of the week) or by the next time for a progress report, which could be Friday; "next week" is not as definite as "the next class".
Thinking about it, I agree. "Read both books before next class" should not be considered correct.
Compare "what are you doing before next class?" to "what are you doing before math class?" In this case "next" is a label for the type of class, just like "math" is a label for the type of class. "What are you doing before THE math class?" would be extremely awkward.
Since it is unlikely that someone could read two books before the class after this one, the focus is on THE next class of this type, not next class in succession. Of course, this is open to debate. In other words, what is being said is "read both books before THE next time we convene."
Accusative feminine (it could also be nominative, but not in this sentence). Masculine form is «о́ба». They look strange because they were dual in the past; now, we don't have dual number, but this word retained its old ending so it looks different from other pronouns.
I think 'by' is more appropriate here because the default meaning, without other context, is that the books must be read by the next class, not necessarily just before the class, e.g. 10 minutes before the class. Stretching has to be done right before the exercise, not at any other time, but books can be read at any time, they just need to be read by the next class.