In my experience with automotive electrical, it can be very tricky to figure out, but the first most important thing is ensuring that you have a fully charged, functioning, battery. There are a great many oddities that can be traced back to insufficient battery performance.
If you have not changed your battery in the past 11 years, it's probably time to replace it. If your battery is somewhat new (< 3 years) there's probably something wrong that is damaging it. Even a budget battery should last more than 3 years without critical failure. Personally, I'd probably throw a new battery in and see what happens.
If you have a couple tools, and really wanted to test what was happening, this is what I'd do:
- Test your battery. Simply disconnect it from your Escape's leads and check it with a voltmeter or multi-meter on the VDC setting. General rule of thumb (that I was taught with) is that it a battery falls below 10 VDC it's probably shot. If you have a good multi-meter you can also check the high/low readings over some hours. If your meter is constantly declining while the battery is not connected, then your battery is probably shot, especially if it's declining =< 11 VDC.
- Test with a proper battery. Either fully charge your battery (if it's healthy) using a battery charger, or get a new one to test with. Over a couple days, take readings 3 times a day (morning, noon and night). This will show if there are recharge issues from the alternator. Plus at this point you can do a lot of diagnostics on your system.
Resolve the issue.
a. If you discover that your test battery is losing voltage outside acceptable ranges, now you have to discover why. There are so many parts that could cause too much draw. Even a damaged alarm system (which I've run into in the past) can cause constant drain and cause you battery to be dead every morning after sitting all night. At this point, I'd probably recommend taking it in unless you're very comfortable and confident with automotive electrical systems. You could also come back here and update your post with new information and we can help diagnose it.
b. If your battery is holding charge from 12VDC-14.5VDC you should be OK, although 12VDC should probably only be right after turning the motor and should go back up quickly.
You have to remember with electricity that many relays, capacitors, motors and such will only work within a specific window of power, but they all provide resistance to some degree. Even the wires, cables and conductors provide some resistance. If the voltage drops too low, the amps will not be able to make it to the needed devices in sufficient quantity to operate the device. This is why the starter is usually the first thing that stops working right - it takes huge amperage and if the voltage is too low, the resistance blocks the flow too much resulting is a weak turn, or no turn at all.
As for things that COULD (theoretically) cause your problem, there could be a relay that isn't functioning at ~10VDC but does function at ~12+ VDC which is causing a short. This would make it seem to work at or below 12 VDC, but stop working at or above 12 VDC. Of course once the battery becomes too drained (from shorting) it will release the relay and allow a new charge cycle to begin and the same symptoms to reoccur. That's a shot in the dark, but possible in theory (without knowing your car's system extensively I can't say for sure if that's possibly).