If you're gonna pirate, I really wouldn't recommend ripping audio from Youtube videos. Low bitrate.
But if that's your only option for something, I would recommend a combination of Video DownloadHelper and Yamb for extracting the audio from MP4 videos. If for some reason only the only option is FLV (Flash video) you can also use FLV Extract. I think YouTube has actually converted all videos to MP4 by now, but I find FLV Extract still comes in handy for some other sites.
If your OBS records to *.FLV, then that's a difficult file format to edit. Download FLV Extractor and place it in the same directory as the *.FLV files, and drag n' drop those files onto the program to create an *.MP4 copy, which should be editable.
It is a good idea to keep recording in *.FLV, as it rarely if ever becomes corrupted in the recording process, unlike *.MP4 (which corrupted as recent as today for me upon testing something), so I prefer to change the format only after I have finished recording.
As for budget video editing software, Sony Movie Studio Platinum on Steam is relatively cheap and works really well for me. I can recommend it, I have not used anything since (other than Windows Movie Maker for certain projects).