Hi Flojo ~ I have almost the identical situation. What I discovered is that immediately after an Amex statement comes out, that is when they report to the credit bureaus. Since the date of the statement if fairly predictable, I simply go online a few days before the statement date and pay off the balance. Thus, when the statement comes out, my balance is zero or very low if a charge or two comes through in the interim, but that will have little or no effect on your score. In essence, you are simply making your payment a few days early - no big deal. I put an automatic reminder on my calendar to be sure I don't miss. Hope that helps.
Starting Score: TU: 821/EQ: 800
Flojo:
Don't forget the catch-22. If you don't carry a balance in any of your credit cards, your score will be downgraded for not using credit that is available. I too pay mine off if a bill comes in and thought with that practice my score would go up. Then it was downgraded for not evidencing credit worthiness, or some crap like that. So I use my debit card (credit union, no fees) and charge my AX when I purchase a book or dvd.
I haven't checked my scores lately, but will do so tomorrow. Hang in there; you'll learn the game.
Hauling said:
1. "Surprising. I get an alert when I use my CC for the first time in a while, but it doesn't affect my scores."
2. "I run everything (quite literally everything except my weekly offering at church, which is by check, and the occasional purchase of Girl Scout cookies) through credit cards, and I nearly always have all but one of them reporting $0.
It's true that if they all report $0, my score drops a bit,
3. so I just let one report a balance."
1a. What kind of alert?
2a. That doesn't make sense. How can you run 'everything' through credit cards and have nearly all but one of them report $0?
3a. That's why I said that I only charge a book or dvd to one token credit card, because I violently object if my scores are lowered without a valid reason.
@MarineVietVet wrote:
@LilyBee wrote:Hauling said:
1. "Surprising. I get an alert when I use my CC for the first time in a while, but it doesn't affect my scores."
2. "I run everything (quite literally everything except my weekly offering at church, which is by check, and the occasional purchase of Girl Scout cookies) through credit cards, and I nearly always have all but one of them reporting $0.
It's true that if they all report $0, my score drops a bit,
3. so I just let one report a balance."
1a. What kind of alert?
2a. That doesn't make sense. How can you run 'everything' through credit cards and have nearly all but one of them report $0?
3a. That's why I said that I only charge a book or dvd to one token credit card, because I violently object if my scores are lowered without a valid reason.
I'll answer 2a. You pay the balance on all but one card before the statements post so that a zero balance is what is reported.
Yup. The name of the game is control your reported util. You can use your cards as much as you want. Just figure out when they report, and what figure they report. In almost all cases nowadays, CCC's report the night that your statement drops, and the amount that they report is the amount that shows as due. If you pay it off before the statement drops (= posts), then the amount due on the statement is $0, and that's what gets sent to the credit bureaus.
1. "Surprising. I get an alert when I use my CC for the first time in a while, but it doesn't affect my scores."
1a. What kind of alert?
A Scorewatch alert. One of the flags that you can choose for alerts is activity on a card that hasn't been used in a while. It's actually a security alert, in case someone has gotten access to your card and started using it. I used to think when I got these alerts that the usage had caused a score change, but there wasn't one, or it was coincidental. The alert was merely to say, Hey, this card is getting used again for the first time in a good while, and were you aware of it?
2. "I run everything (quite literally everything except my weekly offering at church, which is by check, and the occasional purchase of Girl Scout cookies) through credit cards, and I nearly always have all but one of them reporting $0.
2a. That doesn't make sense. How can you run 'everything' through credit cards and have nearly all but one of them report $0?
As MVV summarized above, and as I then blathered about above. 
I regard my CC's as slightly slow debit cards. I don't use them unless I have the money in the bank to pay them off, or money coming in the next paycheck that's already budgeted for that purchase. That's how I can run a ton of money through rewards cards and get cash back. I love my 6% cash back on groceries and 5% cash back on gas, and I do NOT pay interest, and the cards report $0 every month, because I pay them off the day before the next statement drops.
3. It's true that if they all report $0, my score drops a bit, so I just let one report a balance.
3a. That's why I said that I only charge a book or dvd to one token credit card, because I violently object if my scores are lowered without a valid reason.
And that does work when you allow cards to report a balance, but it certainly cripples your use of credit. Try paying before the statement drops, instead of after, if you'd like to use your cards as much as you like without score penalties. Since for most people, having all cards report $0 results in a score drop, I do let one report a balance, and then I pay it off before the due date. (It's scary-easy to forget to pay it off, btw, so you do have to be alert about that.)
I started a new job recently, and with all the accompanying doo-dah and distractions, I let quite a few cards report balances. I never got anywhere close to being late, but just from having the balances report, my scores plummeted. 
Now that the dust has settled, I'm back into my routine of paying the day before the statement date.
Again, I will stress that this isn't necessary, unless you've gotten hooked on maximizing your scores. I can think of worse vices, though.