I'm no expert here, so please forgive the question if it's too vague: I know that a changing electric field can give rise to a magnetic field, and I believe it is correct to say that static magnetic fields arise due to the changing electric fields found in electrons (right?) That being the case, is it possible to create a static magnetic field in the absence of any electric field, or are the two just manifestations of the exact same thing?
The electric and magnetic fields are manifestations of the same thing, the electromagnetic field, and what you call magnetic/electric is observer dependent. For instance, if you had a charge at rest in your hand, you would feel an electric field but no magnetic field, whereas an observer that was passing by in a car would feel a different electric field to yours, in addition to a magnetic field.
Nevertheless, this does not forbid you of having a magnetic field and no electric field. For example, this is the situation when charges move through a wire. You do not feel any electric field away from it (because the wire is neutral, there are as many electrons as protons). However, there is clearly a magnetic field created by the moving electrons, which will be static if these move at constant speed (this can ideally be done just connecting a constant voltage between the ends of the wire).
However, it is true that if you have a non zero total charge, then it is not possible not to feel any electric field. This is because it would violate Gauss' Law of electromagnetism (which tells you that the the total charge enclosed by a surface must be equal to the flux of the electric field through that surface, so total charge different from zero basically implies electric field different from zero).