The problem is not in the page style but in paragraph styles. There are various styles for footers and headers (look at the list in the style side pane). They all descend from Header and Footer which define various parameters shared by all these styles.
I don’t know the technical details but I guess that when you create a document, the default paper format (A4, Letter, …) and margins are used to initialise tab stops in Header & Footer: one at centre, one at right.
Settings in styles are static. This means that they don’t change (automatically) once they are initialised.
In any page, by default, a header is styled Header, using the configuration of the style. Obviously, since it was initialised with portrait orientation, distances are incorrect for landscape.
The solution is to create a paragraph style (or a set of) fit for landscape and to apply it on the landscape header(s). Note the possible plural form because the same style can be applied over header in several page styles when the print width is the same (paper width minus margins).