Regardless of the legality it will be infinitely easier for you to travel if you have a French passport. I suggest contacting your local consulate with whatever documentation you have (birth certificate of your and preferably proof of citizenship of your parents).
You can certainly enter France with a US passport and try to sort it out locally while you are in France. Just make sure you carry all documentation that you may need. Once you have "easy" documentation of your French nationality, you need to switch your arrival record from a US passport to the French passport. One easy way to do this, is to simply take a day trip to the UK (or any non-Schengen country), exit on your US passport and re-enter with your brand new French one.
Provided you meet all the criteria to be a French citizen, it's probably not illegal for you to stay for more than 90 days, but if you get checked after 90 days with only your US passport, you will have a LOT of explaining to do. Passport checks at Schengen borders are rare but they do happen occasionally and you will have to show it when you leave Schengen. Hotels may also want to see your passport when you check in.
Recommendation: sort it your before you travel.