I'm not sure you can claim both at the same time, i think if you claim ESA then your partner will have to go on that claim instead of being in JSA.
Also depends if you're on contributary based or income based.
Contributory ESA [footnote 3] will contain only a basic allowance during the assessment phase, which is paid at a personal rate for the claimant only. There is no couple rate or any dependent or age additions like there is in IB. But where both members of a couple are entitled to contributory ESA they will each get a basic allowance:
Single claimant aged 25 or over: £60.50
Single claimant aged under 25: £47.95
Income-related ESA [footnote 4] is calculated in a similar way to IS. It has an applicable amount made up of allowances for single claimants and couples, premiums, and eligible housing costs:
Single claimant aged 25 or over: £60.50
Single claimant aged under 25: £47.95
Couple, both aged 18 or over: £94.95
After 13 weeks you'll be assessed and either put in the work related activity group or the support group (or deemed fit to work in which case you'll have to go bac on JSA)