You're going to need your original SIM in the phone to use your US phone number. That also means using your US phone service in Europe. Some phones are "dual SIM" now - if you have such a phone, you could buy a SIM in Europe and mostly use that for data but also have your US SIM available for calls.
What kind of US phone service do you have now? Do they have an international roaming plan?
You can certainly buy a French SIM when you get there. The Orange Holiday SIM that you can buy in France will work in Switzerland as well, but I think it's good for only two weeks before you have to add more credit to it.
I've used my Dutch Vodafone SIM on my last three trips to Europe - bought the SIM on eBay, haven't actually used it in the Netherlands, but I have used it in France, Italy, Slovenia, Spain, and Portugal. (It should also work in Switzerland.) An advantage over the Orange SIM is that it would probably be cheaper, especially if your trip will be for longer than two weeks. But it is not a great choice for calls - it's mostly good for data. You can always use a app to make voice calls to US numbers e..g. Google Voice, which lets you make free calls to US phone numbers, even landlines, from either WiFi or using mobile data. But you can't receive calls as far as I know; Google Voice gives you a US phone number, so people could text you and you could call them back - or they can leave you a voicemail and you can call them back, for free.