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If the turn signals work on the trailer, the issue shouldn't be the wiring for the Mountaineer, if you're truly using a factory wired trailer plug. The only relay for the trailer adapter in your 2000 Mountaineer, per the factory wiring diagram, is for the trailer parking lights. The turn signals and brake lights share a signal to the trailer plug and the switching for either turn signals or stop lights is performed within the Multi-Function Switch (MFS).
Have you traced the wiring back from the plug to confirm this is factory wired? If the factory wiring is utilized, the harness should be plugged in to a round connector under the driver's side taillight. If it goes in to an existing wire harness, it's probably an aftermarket install and it may not be wired correctly (wired to turn signal circuit which is independent from brake lights).
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