Now this may seem like the silliest question around, but please answer it nonetheless.

Is there any way to cut the video to the next clip, but keep the audio from the previous clip overlapping onto the new one. An example is someone talking, its a close-up on their face, and then it cuts to the person they are talking to, but the persons voice is continuous without the use of a manual voice-over

any help would be greatly appreciated. 

Welcome to the forum

To do this, you should work in timeline mode. You can lock the video track and edit the audio, or vica versa.

If you have a manual, look at the section on J-cuts and L-cuts.

What version of Studio do you have?