Airfares to Germany this Sept.

Use Skyscanner.

I would observe that Lufthansa is a partner with United (and has code shares on United aircraft, and vice versa). I have usually found that two-segment flights involving Lufthansa deliver me to a non-hub city with little delay and considerable comfort, at little or no additional cost. That's using the United website. If you are a Mileage Plus member, award travel for two-segment flights is much easier than single-segment flights. Don't forget to save precious time, and train money, with Open-Jaw booking.

Here is quote from yesterday's New York Times:

The Airlines Reporting Corporation (ARC), which provides air travel intelligence, released a report last summer on one-way tickets that concluded “the long-held belief that it is better to purchase round-trip tickets whenever possible to get the best fares is simply no longer true.”

https://nyti.ms/2N4pkmr

I guess from the post that you're never a Business Class traveler, but I can report that Lufthansa Business Class is nicer than United Polaris Business Class.

I think most people would suggest that you start your study with a consolidating website like Kayak, but I also use Expedia because they have a deal with the AARP. But many of their search results are with resellers and consolidators, which are of less interest to me. I like to have a seat assignment when I pay my money. Personal preference. The NY Times recommeneded Google Flights yesterday, but I have yet to try it.

How many days are you staying?
Are you visiting only Germany or is your plan to land in Munich and go to other countries?

I see that Delta has round trip fares from Houston to Zurich for 14 days starting at $480.
For 7 days they start at $477.
You can them fly/train to Munich from Zurich.
I used matrix.itasoftware.com to check a whole month.

I like to use Google Flights for research. There’s a lot of flexibility. Just enter a departure airport and dates, for the destination you can enter a specific city, a country or “Europe”. It provides results for non-stop flights and connecting flights. I haven’t been successful in trying to get results for open jaw flights.

Try Condor from Austin direct to Frankfurt. Last September it was the cheapest way to get to Germany for many in my Oktoberfest group.

DJ

Edit: quick check shows AUS to MUC Sep 17 returning Sep 24 for $840 on Condor.

if you use Condor, they offer Rail&Fly. For 29.90 EUR, you can take any train from Frankfurt airport to anywhere in Germany included with your ticket. I see no mention of it on the US Condor website, but it is on the German site. Use Google translate.
https://www.condor.com/de/flug-vorbereiten/am-flughafen/railfly.jsp?ns_linkname=affilinet&utm_term=v01MTQyMTExMTExMTExMTExMTEwMTQyNzAyMDAwMDAwNjE1MzA1NTY1NDdjM2llbGRlNTc2NDYxLWRlNTc2NDYx&utm_medium=Affiliates&utm_campaign=affilinet&utm_source=576461

Wow! Thanks for all the suggestions. I will look into them. My trip is primarily to Germany, but will also include Swiss Rail for several days. Looking to go for 4 wks.