I purchase everything with my credit cards. Can I use the total I charged for the year on my credit cards to calculate the sales taxes I paid?

Yes, The IRS allows you to deduct actual sales tax paid or a standard formula. 

If you have all of your receipts for sales tax paid during the year, you can deduct the total sales tax you paid. But if you are audited (this is not an audit trigger unless it is abnormally large compared to your income), you will be required to produce all of those receipts.

So can I just back out sales tax amount from the total of all my credit card statements for the year?

Yes. Assuming everything is taxed at the same rate and there are no items exempt from sales tax.

paying the same % sales tax on everything you put on your CC is highly unlikely. so if you didn't, you would need to go through every receipt.