I found a solution. By searching on the above phrase, I found the following thread:
https://gitlab.freedesktop.org/pipewire/pipewire/-/issues/2739
In it, it was suggested to run the command:
pw-metadata -n settings 0 clock.force-quantum 512
That fixed the problem. I tried 1024, which also worked. I then tried 256, and while it worked, I had momentary little gaps in the audio at random times. So I decided to stick with 512. The person who suggested that solution said that if you have to go to 1024, then it’s possible that your machine just can’t handle the “required latency”. So, I suspect that wine adds latency, and that’s why non-wine games were working fine before making this change.
I hope this helps anybody else who might run into this problem.
Thanks for pointing me in the right direction @marel