"Moving house" vs "moving houses"

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.

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

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.

In our experience, people refer to a change of residence as "The move houses." However, the traditional way people express about service sometimes says "I plan to move house next month" and with this is implicit that is speaking from one direction to another.

Taking the analogy from "changing places", it should be said "move houses" Puime

As an older American who has lived in the Mid-West, South and California, I have never heard or read this in American material. I first came across "move house" in British ESL/EFL material, where it is already introduced in pre-intermediate material, e.g., Fun for Flyers from Cambridge Books.

I would simply say, "We're moving" or "We're moving to a new place/apartment."

Looking at the Goolge ngram viewer, I found that "move house" and "moving house" are in fact present in American-English books, but at about 1/6 the frequency found in the British material. Interestingly, besides the surge of the phrases in the later 20th century, there was a lesser surge in the 1820s, in which the US and British frequencies were about the same but with the British activity starting later and carrying on longer.