Naomi
Based on your post I can see no reason why you would have to claim Universal Credit but it is possible that you would be better off doing so.
If you stay on ESA then there is a £20/week disregard but after that your partner's income will be deducted £ for £ from your ESA.
Under UC the LCWRA rate (the same as ESA SG) is higher and rather than £ for £ their earnings will be deducted at 63p for each £.
Try putting your information into one of the online benefit calculators
www.gov.uk/benefits-calculators Gordon