An HD60 (non-S) is a USB 2.0 device; it will always have around a 2-second capture delay on both audio and video, which cannot be fixed. You can add an async delay to your other sources (like your webcam, alerts, etc) to sync your mic/cam to the gameplay so you aren't reacting to things that haven't happened yet on-stream, but the capture and audio will always be 2 seconds late for you.
You can plug headphones into the Switch, but that kills the audio going to the capture card.
You can use an HDMI splitter and have the video/audio going to a TV as well as the capture card.
You can plug a dubbing cable from the Switch to your computer's Line In, and set the line in device in Sound->Recording to 'listen to this device' which will play back the audio from it over your desktop speakers, and have OBS capture the game audio on the desktop channel, but keeping game audio/video in sync can be difficult to impossible at that point.
The real fix is to sell off the HD60 to someone else, and buy a USB 3.0 or PCIe capture device instead. This is otherwise not a fixable issue, with the HD60.