You might get more responses in the Air Travel forum.

Use Google flights. It's the cleanest look-see at available flights and prices even though you may not be able to book directly from there but may have to go to the carrier website.

1) Check for prices on Tuesday or Wednesday as they are many times the best days to book at the lowest prices.

2) Be flexible on what days you travel and to which airports. Run all the acceptable scenarios and you'll find some are just lower. Frankfurt has a lot of direct flights to USA but direct is not always the cheapest.

3) Be realistic on what you will pay. For me r/t Europe flights for $1000-$1200 are very reasonable, $1200-$1500 is too much and $800-$1000 is a steal. But I fly from the US West which may be a little more than TX.

4) Book at the right time. Three months out is usually the cheapest but not always. Sometimes it pays to wait but I would be checking regularly starting at 4 months out. Thats when prices seem to see movement. Before that they don't change much usually.

I would avoid the Ryan Air connection unless you can save significant $. The hassle factor of making connections work with discount airlines is too high for just a couple hundred bucks savings.

I am flying r/t to Frankfurt in June for $1,050 direct on Condor Airlines. Good price, so so carrier. But I can put up with minimal leg room and will sleep at least half of the 11 hr flight so I'm happy!
Good luck!