Well, what I did was only do the exercises on the last day of that lesson. Re-reading the whole lesson if you screw up seems more like punishment than reinforcement. Rather, I don't worry if I screw up; I just try to understand how I screwed up, and why the answer was what it was. Then I move on. And yes, you are remembering the answers, that's just how your brain works.
What's nice about that book is that it recycles some vocab words and grammar in the following dialogues. That way, you're less likely to forget stuff, and you'll better understand it. A perfect example for me would be relative clauses (The fat cat that was sleeping on the couch). Seemed impossibly hard to wrap my mind around Japanese relative clauses at first; not too many lessons later, it became really simple to me. Personal recommendation though: flash card the vocab, or whatever you do to remember it. The "recycle" method will make sure you have all the important grammar down, but if you're anything like me, you won't remember much vocab just through the exercises and recycling. Put some extra work into remembering those words. It'll boost your ego too, once you get to the reviews that come every 5 lessons. I crashed and burned through those solely because I didn't remember the vocab.