Travelling?

Since you entered Switzerland on your US passport, you have to abide by visa regulations for US citizens. That means you need to leave the Schengen Zone entirely, not just cross into France or Germany on your US passport. You have used up (or about to use up) your maximum time in Schengen, and have to depart entirely. Besides, within Schengen, they do not stamp passports. Get out of Schengen Zone entirely, or you have a problem - Really Big Problem!

Go to UK, which is not part of Schengen, or return to US. Return to Switzerland (or anywhere in the Schengen Zone) on your new Swiss passport.

I have renewed it. I made sure to do it asap when I got here. My issue was having the other passport stamped... I didn't know how to deal with the situation as I traveled on a passport that was up to date. Thanks!

You cannot get a stamp in your passport when travelling from one Schengen country to another. You have to leave the Schengen area. If you think about it for two seconds, I'm sure you can work out why.

Why would you not renew your Swiss passport? If you have been in that country long enough to trigger the time limit on your US passport, you have certainly been there long enough to renew.

Once you have a valid EEA passport, exit Schengen and re-enter the zone. Most people fly to London.

UPDATE: The Swiss authorities may not mind how you entered. Go to the Cantonal Amt für Migration and ask for advice. Take both passports to them. They may be willing to stamp you out.

If you are traveling on your US passport you are a US citizen and that is the only thing that matters, if you wish to be treated as a citizen of another country then you travel on a valid whatever EU country passport