I assume that you're using a WS2000 (or maybe WS1000) Spektrum USB receiver dongle as well.
What you need to do is map the DXS radio channels (provided through the WS2000) to RealFlight's virtual radio channels. RF uses different channel numbers than on Spektrum radios.
I have a writeup
here that was done with a DX9 and a WS2000. You can't change your DXS in the same way and you really don't need to, but that document will still cover the process for mapping the channels in the RealFlight Controller Profile to get what you need. To solve the flight mode issue, you want to get your flight mode switch associated with RealFlight Channel 8 "Mode". It's all covered in that writeup. After reading the intro on page 1, you can jump ahead to the bottom of page 2 which gets into the changes on the RF side. If you aren't sure which channel is which on the DXS, you can just select the input to change and flip the right switch on your DXS.