As far as the practical solution goes, the following may make your CC life easier.
Call your credit card companies and ask them to move the start of your billing cycle to (say) the 25th of the month. Most card companies will do this for you. Then just make the 1st of the month or thereabouts the day that you pay bills and review charges. Get it all done on one day in about 20 minutes. If you are willing to, add auto-pay or auto-draft as a backup to make sure there's no possible way you could ever miss a payment. Personally I set them all up to pay in full since that forces me to always keep a couple grand in my checking, but not everyone likes that.
Now to address the issue that the OP brought up. Divide your cards into a few categories. First there are the cards that you are sure you will use at least every other month. Those you don't have to worry about. The rest are the cards that might possibly be unused for at least three months at a stretch. Of those see if there is some kind of recurring expense that you are sure you would charge anyway. Netflix is a favorite, so is Amazon Subscribe & Save. Now those cards are taken care of. Sock drawer the rest. Make it a rule to take them out once a quarter: Jan 1, April 1, July 1, Oct 1. Use them for something small that you were going to get anyway: one per item at the grocery works. Done. I am guessing that the problem our OP described (assuming Bigfoot is real) doesn't kick in until at least three zero balances in a row occur, so quarterly would be enough.
As far as the general problem of maximizing score benefit by CC payments, I personally think it is a bad idea to worry about it. I think the thing to do is to always pay in full if you can, and if you can't work toward that as your main goal. Don't worry about your utilization ratio, just let the amounts report as they would naturally. If you have a high utilization some month, your score will drop. It doesn't matter. The "bad" effect will disappear the next month. In fact, as many have argued, there's a long term advantage to you for your CCCs to see that you often have balances and always pay them in full. Now of course, if you need your score to be maximally good some month, just pay all the cards down to $0 except one and leave the remaining card < 9%, with a total U of maybe 1-2%. But doing that every single month may be more worry producing than necessary.
The idea of permitting your U% to go up a lot some month (with a score hit) may freak you out too much to be worth it, even though you know in your head that the hit is very temporary. In that case just make a couple payments per month on the cards that you charge the most on and keep your total U at 1-5%.
In summary, use your cards for the stuff you need, let them report naturally, keep your U% and spending low, and you'll be fine, Once in a great while you can squeeze a few extra points with the "All cards $0 except one" rule but that's only when your really need a few extra points and it should be done rarely.
PS. It's also worth observing that, if you are genuinely anxious about improving your score, don't get any more cards (since it looks like you have 7). The best thing you can do is let the ones you have age and have a perfect payment history on them. It's entirely possible to score in the 830s with three credit cards.
