I am currently using a Sony DXC-D30 camera for a project, but there is an issue with the auto white balance.

At first I didn't notice it, but the when I was looking back at footage of an interview you could see the white balance "sweeping" from balanced to unbalanced. It's as if the camera is getting confused with the lighting conditions, but the problem persists when you take it outside. With the viewfinder being black and white I have no idea how the colors are behaving when shooting, and it's not always convenient to manual white balance.

Does anyone know what may be causing this or how to fix it? It's really distracting and makes the footage look awful!

Auto white on a camera of this type really is not the way to go. I know it can be a PITA to constantly WB but on the DXC30/50 AW, just like the JVC5000/5100 and other cams of this nature just doesn't work well.
What I've done is to set a WB on 1 area and set that to A then set WB to another area in the opposite direction and set that to B and while it can have a jarring effect if the WB is tat much different it's the fastest and easiest method that I have found for doing run and gun stuff.
Obviously it may not be the 100% best method but I've never been able to come up with another way.

Does anyone know what may be causing this or how to fix it? It's really distracting and makes the footage look awful!

There is only one way to fix this and that's to manually whitebalance each time the light conditions change. If you want to get it right never let the camera decide, a camera can get quickly confused, even in constant light conditions, f.i. if you would film an interview and pan from one person with a white shirt to a person wearing a yellow or green shirt your camera already could get confused.