What form do I need

The form is the ESA3 form and the reason this has been sent is because of you moving in with your partner. However, if all of your ESA is Income Related because your partner is working more than 24 hours per week then there would be no entitlement to Income Related ESA so I'm confused why they have sent that form. You can see the form here.

https://gmwrag.wordpress.com/wp-content/uploads/2016/05/esa3.pdf

You say you are about to move in so once you do move in you will need to ring ESA again and tell them you've moved in with your partner. I realise you've already done that but they can't action changes before that change actually happens.

If any part of it is contributions based then this will continue (£138.20/week) as it's not affected by a partner working.

Have you checked entitlement to Universal Credit by using a benefits calculator? As its means tested you would need to claim as a couple and your joint circumstances will be taken into consideration.

As you're in the Support Group for ESA you'll be entitled to the LCWRA element from the start of your claim. You and your partner will also be entitled to the work allowance, which means some of your partners earnings will be ignored before any deductions apply. Therefore there maybe some entitlement but it will depend on circumstances. Benefit calculator here.

https://www.entitledto.co.uk/benefits-calculator