Is it possible that your tow vehicle does not shut off power to the 7-pin when turned off? If shorted, this would be a power source for the lights when the vehicle is shut off. This does not explain the signal light not working.
I would first clean and check the connections on the 7-pin connector on both the tow and trailer sides, as well as the connections and bulbs (assuming in 2007 the did not use LED yet). Other than that, it pretty much has to be something in the wiring.
2017 Escape 5.0 TA
2015 Ford F150 Lariat 3.5L EcoBoost
2009 Escape 19 (previous)
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