Hi David,
Quick heads up first, you cannot turn off WASAPI itself.
It is the Windows Audio Session API, not a switch. What breaks Snipping Tool recording is usually an app grabbing your sound device in Exclusive mode. The reliable way to get Snipping Tool working is to take your default playback and recording devices out of Exclusive mode so Windows uses shared mode. There is no global toggle for all devices, so you only need to change the two devices you actually use, not all six.
Do this now using the classic sound panel:
Press Win+R, type mmsys.cpl, then press Enter. In the Playback tab, click the device with the green check mark, choose Properties, open the Advanced tab, and clear both "Allow applications to take exclusive control of this device" and "Give exclusive mode applications priority". Click Apply and OK. Switch to the Recording tab, repeat the same two checkboxes for the default microphone, then close the dialog and start a short test recording with Snipping Tool.
If you intend to include system audio in your clip, make sure Snipping Tool has system audio enabled in its own settings before you hit Record. Open Snipping Tool, select the three dots, open Settings, and under Screen recording turn on "Include system audio by default". You can also toggle mic and system audio from the toolbar on the recording screen.
Tell me how that goes. If Snipping Tool still complains about Exclusive mode after these changes, I will walk you through the next step.