Ooooh it might be because of that. To be honest, I'm not sure whether you should actually do a jump cooldown😅, don't know if its a stylistic choice but most platformers dont do it because they want their jump to feel responsive.
Since you're using Unity I'd suggest you look into Cinemachine and how to setup multiple Virtual Cameras, that's what I did for my game and that should help a lot for the camera!