Holy Jesus... It looks like after 2 years, I accidentally found a solution to my problem.
I do not know why, but apparently in my case the problem was - the USB port ... It's shocking.
Apparently, I did not understand very well what the problem was. The problem was not in the capture card, but in the audio monitoring device and how it works in the OBS. I poorly explained in my first post how my setup works and this is a very important detail. I use obs to listen to the sound from the Console / PC, which the capture card outputs and stream at the same time. And it is this audio that gradually begins to lag behind the picture.
I use a Blue Yeti microphone, and I listen through it to hear my own voice and surroundings, because I don't like being "deaf" when playing. That is, it is both a microphone and a sound output device at the same time, there is a headphone jack on the microphone to mix PC sound with what the microphone captures. I decided to try my 2-PC setup again when Сyberpunk 2077 came out ... And suddenly for myself I found that there was no problem. I played CP2077 for 12 hrs straight and streamed it and had no problems whatsoever. I thought maybe the problem was just fixed in one of the patches (because, as I wrote earlier, this problem applies only to OBS), but the problem reappeared after a while, after I moved my second PC to a more convenient place. I was confused, began to think, what did I do, that the problem reappeared... In the end, I plugged my USB microphone into a different port and the problem went away. Perhaps the problem is in the chipset, some USB ports in the device manager are determined by the controller from ASmedia, others from AMD. If you are also listening to audio from OBS through a monitoring device and you have a gradual audio delay issue. Try plugging your audio monitoring device into a different USB port. It helped me personally.
P.S. Thanks everybody for your suggestions and willingness to help. I really appreciate it.