I've had it confirmed this morning at Citizens Advice that I can apply for Carers Allowance whilst on ESA.
First bit of good news for a while and thanks to this forum for originally saying this is possible.
If I get it, I wont be paid it, but will just be awarded it. My plan is that the CA will kick in when/if they throw me off ESA now that I am having to the work capability assessment again (which seems likely looking at the horrendous scoring for anxiety/agoraphobia type stuff).
Oddly, they failed to mention to be careful that you don't inadvertently show that you are doing carering duties that you shouldn't be capable of because of your own health.