The cuts they are talking about in the house are much worse than limiting disaster relief funding, if you can wrap your head around it.
The cuts they are proposing are t the agency itself, and will slash it's ability to plan, prepare, coordinate and provide resources on the ground both before and during an actual emergency. The agency is the backbone of relief efforts.
The $6B congress is talking about is to pay obligations relating to insurance claims, road repairs to federal highways, power, sewer and water infrastructure, and to provide temporary housing. It's all after disaster money.
Without an agency to plan and coordinate, disasters will be much worse, and many more lives will be lost.
They are shredding the infrastructure of government and putting the money into their own pockets.