If you have about a hundred bucks, Camtasia is an excellent product. Video editing suite and also a screen recorder, records exactly what you see on a monitor and also system sound and can be set to also record your mic as well for voice over work. A lot of companies use the software for training and instructional videos. A lot easier to pick up than Adobe products but less powerful for post production. 1 time license fee too so SaaS can fuck off.
I just bought it for the screen recorder for some projects I do and it has worked well. Your use case would work but there are free options out there that may be more your style. My $.02 FWIW.