I have my docked Switch which has an HDMI cable running to an external Elgato HD60 capture card. The capture card is connected to my monitor via HDMI, which does not have built in speakers or an audio jack. The card then has a USB connection to my streaming PC (duh).

For streaming, I have OBS using Game Capture Software as a vidoe capture source. This is important (I think).

Since I don't have speakers on my monitor, and the docked switch does not send audio through HDMI if you have headphones in, I can't get game audio to myself while streaming (it captures on stream and in recordings just fine). So, we go to the computer to capture the audio. Elgato's Sound Capture won't work unless I have Game Capture HD open, which means OBS won't work. If I tell OBS to send audio from the capture to my speakers, I don't get it in the recording.

Does anyone know how to set it up so that I get game audio in my recording and also sent to my headphones while I play?

You could use an Y cable to split the audio from the headphone jack of the switch, then use one for your headphones and one to an audio input on your streaming PC, simply have OBS delay the audio output by as much as your external card's video feed is delayed.

Example of Y splitter cable on amazon

(with only reading the title I thought you were having issues with audio relating to Surround Sound, on my card that leads to BotW's voices to be missing from cutscenes.)