ESA for Students

Hi Everyone, I have a query about ESA as I have read a lot of conflicting information and I am not sure where I stand. Both me and my partner are fulltime students but he is also my unofficial carer. He works on weekends but is at risk of getting fired due to the number of days off he had to take because of me We found out about ESA but apparently I cannot make a direct request for it because my area has the full range of Universal Credit, so we applied for that and had the first interview a few days ago, but the woman told us we are most likely not gonna get any help due to our student status, but we are gonna have other interviews. When we mentioned to her the ESA she said that ESA is not available for students...is that really the case? NUS seems to say otherwise and honestly I don't know where else to look, we are struggling a lot and if he is gonna lose his job we are gonna left with no money at all. I am getting PIP but it's not enough to cover monthly expeses. Any advice would be highly appreciated. Thank you