JonnySkell
The DWP are not treating you as a couple at this time, your ESA and PIP are being treated as individual claims, so your moving out will not effect either claim.
To be paid a Premium for ESA, one of you has to make a claim for Income Related ESA, this will be paid as a top-up to their ESA. This will be affected by whether you are together or not, the DWP will have to decide whether you remain a couple or not.
I would get face to face advice, before making any changes, a trained advisor can work out what you would receive while together and apart, they can also help you if there are problems with you living apart.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems