Good advice from mathschick. Also do run in the week but if you can try to mix it up to keep enjoyable - 10k every other day sounds like too big (and I'm also thinking unneccessary) a challenge right now, but you have already found that out yourself.
It is a bit confusing because I always thought you recover from injury and rest during days off. But now I gather runners tend to only rest when they are suffering with an injury, and recover when easing up a bit after consecutive weeks of hard training. Perhaps rest days or longer periods of not running don't come into it when you are fit? But experts advice I've read is to always respect days asigned for rest, while recovery weeks (running on average less than you're used to) will ideally finish with a race of a shorter distance than you're training for.
The training plans I've followed suggest you take it easier than usual in the days leading up to race day, so you're feeling as fresh as can be considering you've trained hard for x number of weeks before..