HelloCustomer The first thing you want to do is a load test on the battery, take it out and bring it to autozone, they usually do this kind of test for free, you can still have enough voltage to operate components, but a battery that has an internal fault will have similar symptoms when a greater load is put on it. If the battery checks out ok, I found a technical service bullitin that might apply, you might need to get the vehicle to a dealership to have the powertrain control module reprogrammed. (see below)
Some 2004-2005 Taurus/Sable vehicles built before 8/27/2004 equipped with a 3.0L 4-valve engine may exhibit a no crank, battery discharged condition.
Reprogram the powertrain control module (PCM) to the latest calibration using WDS release B34.7 or higher. Calibration files may also be obtained at www.motorcraft.com. If a charging concern is still present after PCM reprogramming, perform normal diagnostics.
Eligible Under Provisions Of New Vehicle Limited Warranty Coverage