From your screen recording, I can see that your source videos are variable frame rate. Media players can usually do fine with VFR. But video editors do not work well with it because it is difficult to seek.
Your best approach would be to configure OBS to record in constant frame rate. See this post and other similar posts on this forum:
Another option is to click “Convert…” and convert your files. The converter in Shotcut will convert your file to constant frame rate and then replace your source file with the converted file in the project. It takes some time and it costs some disk space. But it usually makes the files work.