How to create an overcast weather?

Need some help trying to figure out how to create an overcast weather. Is it possible only using an HDRI or changing the skysphere?

the skylight is what is commonly used to simulate an even tint of light across all assets - similar to what you get in an overcast day.

I would suggest you give using that a try and come back with any specific questions you may have.

For shadow casting purposes you may still wish to combine it with a light source.

For changing the basic skysphere you can create an instance and replace the cloud material. Or as you mentioned utilize a cube map.