If your doctors letters have a different address to your actual address, is that a problem?
For example imagine you want all your letters redirected to your family as they read your mail and support you, so you request doctors to send letters there.
Likewise can you apply for pip or esa using a correspondence address/or family address as you want them to deal with any correspondence.
If the DWP aren't aware of the reasons why two different addresses appear on your information, it might be picked up and cause you an issue. Just make sure they know.
BIS
Nothing on this board constitutes legal advice - always consult a professional about specific problems