If I'm in Turkey on a US passport, but have a Turkish passport, do I need to leave the country to stamp out on the US passport and not overstay my visa? Assuming I'm a citizen of both countries (and have always been a Turkish citizen with a Turkish passport).
Türkiye allows it citizens, who are dual citizens, to enter on their other passport.
So that is not the problem.
Inside Türkiye you should be treated as a turkish citizen, but if you didn't state that when you applied for the eVisa they won't know that.
Showing you turkish passport when leaving should resolve any problems about the visa having expired.
Turkish nationality law - Dual citizenship - Wikipedia
Dual nationals are not required to use a Turkish passport to enter and leave Turkey; it is permitted to travel with a valid foreign passport (or national ID card for some nationals) and the Turkish National ID card.