When selling products that require quarterly excise tax payments, how do you accept payments without paying taxes on the product until it is actually completed and shipped? In other words, I want to avoid paying taxes in quarter 1 if the product will not fully be paid for until quarter 2 or 3 and shipped then.

You do NOT.

Most businesses operate on the "cash" method of accounting, which mean the payment is recorded when made, so that is when the taxes are due.