Rental house smells musty moldy when turn ac on. What could be checked for?

Despite the fact I’m living in a gross old house with a probably semi wet basement and should truly move the hell out… I still am interested to know what could be checked with the hvac on the off chance it could be less gross somehow. I believe the hvac and ducts were replaced like within the last few years. I’m pretty sensitive to mold and so today we finally put the ac on as it’s been getting hot AF and humid here and I instantly could smell it. I didn’t feel like it went away until the air was later turned off. Is there anything that could be making this worse besides just I live in a gross old house and it’s probably pulling air from the basement ? Or could it be something that could potentially be cleaned and is that in a cost range of something a renter would pay to have done? It’s hard because all houses in the area seem to have some sort of water damage mold issues as it’s a humid hot climate in the summers. Maybe it’s just time to move elsewhere.