There are situations where more than one SW payment can be made at the same time(e.g. One Parent Family Payment and half-rate Jobseeker's Benefit).
OP, as partnership said, one of the qualifying criteria for Jobseeker's Allowance is to be available for, and to be seeking full-time work. This may clash with the requirement for Carer's Allowance where you cannot work for more than 15 hrs pw (with permission from SW). Jobseeker's Allowance would also be means-tested on your partner's income, and the means-test is less generous than that of Carer's Allowance.