How does paying off one of two mortgages affect my credit score?

Credit utilization (and therefore, your credit score) isn't impacted by one's mortgage. I have a current mortgage and saw a zero utilization the one month that I paid my credit cards in full just before the bill was cut. What will change is your borrowing ability. Your debt to income will drop and the payment you used to make will be available to service a future debt if you wish.

Note - obviously, a regularly paid on time mortgage helps one's score and late payments will hurt. I am addressing just the 'utilization' portion which is what i think the OP is asking. The credit report, strangely enough, does not take income into account. The focus instead is on utilization.