Jeff
The first question is whether you ESA is Income Related or Contribution Based or combination of the two?
If any of your ESA is ESA(IR) then you will have to claim as a couple, this will not require a claim for UC. However, you say your partner is working, whilst your ESA will be increased to the couples rate, any income they have will be deducted, although there is a disregard, also, if they work for 24 or more hours a week then your ESA(IR) payments will stop.
There HB may be effected.
I would put your joint information into one of the online benefit calculators to see what you might receive.
www.gov.uk/benefits-calculators Gordon