black2cat
Based on your post I can't see why you cannot claim Carers Allowance?
The claim would be in your name and is not affected by your husband receiving ESA(CB) , although this would not actually be a problem anyway.
Again, as this would be your claim there is no reason to assume it will have any impact on his.
I would recommend that you that you check which benefits you are eligible for on the brief information you have provided your husband would be eligible for at least one of the ESA Premiums if you are eligible for ESA(IR) as well as his existing ESA(CB).
Try one of the online benefit calculators.
www.gov.uk/benefits-calculators Gordon