Covid Travel Restriction Rules for Foreigners Living Within EU

As France and Switzerland are both members of the Schengen area, you can freely cross borders as long as you hold a Schengen visa (which I suppose you do).
In the current situation, of course, there might be added restrictions and/or quarantine requirements. You should inform yourself about those - or ask on the relevant country forums -and keep in mind that they can change at any time without notice.
If you want to hear my personal opinion: I would rather not travel right now!

Contrary to what Beppi wrote, Americans (plus citizens from a handful of other countries like Canada, Australia, New Zealand and South Korea) do NOT get Schengen tourist visas issued. They come visa-free for up to 90 days in a 180 day period. Same time limitations as for a Schengen visa but without needing the actual visa. Crossing the borders one might not get checked - but if so it could be a problem when one has no long term visa or residency permit. But different countries seem to be interpreting the Covid regulations in a different ways; all have their own twist and conditions seem to change constantly. So like Beppi suggested one should look at a government website of the countries involved to see what the current conditions are.

My suggestion would be to travel across Germany to France without going through Switzerland. There are usually no controls at most German/French border crossings but almost always at all crossings in and out of Switzerland.

But the original poster’s statement they formerly left every 90 days for a visa renewal doesn’t make sense. They would not be getting a visa but complying with the time limitations for their status. But one does not leave and then simply come right back in with allowance for another 90 days. They must stay out 90 days before returning. But if one is registered and a residency permit is being processed, then they would not likely have to leave even if going over 90 days. This should or could be clarified with the office doing the residency permit.

I stand corrected about the visa situation - sorry!
But as Tom wrote above, the same rules apply to you as if you had a Schengen visa.