"Moving house" vs "moving houses"

In the UK I think people only ever move house, unless they're talking about more than one household moving at the same time. But in the vernacular, house builders and estate agents (US realtors) might well talk of "moving houses" to mean "selling houses".

This NGram indicates "move house" is the dominant form, but switching between American and British books suggests that tendency is less marked in the US.

If I intend to relocate my belongings to a new residence (be it a standalone home, apartment, dorm room, or work cubicle), then I would say I plan to "move house", or simply "move". If I'm moving a large distance, I might say I'm "moving away".

If I intend to sell many residences (because I am in the real estate business), then I would say I plan to "move some houses."

Alternatively, if I am in the business of physically picking up and hauling around buildings, I might say I plan to "move the house". (I would probably tend to preface this statement with an explanatory remark, like "I'm a housemover".)

With your original sentence structure, I would tend to say "I plan to move next month", or perhaps "I expect to move into a new house next month."

If someone told me "I'm planning to move houses next month", I would probably have a moment of cognitive dissonance thinking that they were in the housemoving business before deciding that they really just meant they were moving their belongings into a new house.

As you are the one moving, perhaps leave the houses out of it. They are staying put.

When I use "move house" I use it as a general analogy to completely move or change from one thing to another, be it my actual house, my cubacle, a shelf in my cabinet, etc.

When I use "move houses" I am specifically talking about changing my place of residence.