How to apply for Carer's Allowance?

I have lung cancer and live with my 2 sons, one is autistic and really doesn't leave the house and the other has had some small strokes resulting in a change of personality.    Neither have applied for Pip because of mistrust of the Dwp and i can't blame them.   I now really need some proper help regarding care, shopping cooking and personal care.     My Autistic son would be the one who would apply or should I say i would apply on his behalf.  My question is.. When i die would his benefits go back tO. ESA or woyld he have to reapply.  If this is the case we  would not be able to apply for carers allowance because it would cause him an inordinate amount of stress and i could not put him in this position.  I have tried to find out answers to questions on line and from Macmillan to no avail.