Based on the info and photos you provided, I am also assuming the black specks are caused by insufficiently rinsed carbon.
The amount of carbon is not the issue. 1.5 ft3 is the usual recommended minimum amount and a greater quantity is always better. In a backwashing system, 1.5 ft3 media would usually be installed in a 10" X 54" tank.
The actual issues are the lack of backwashing, and the amount of media stuffed into a small tank.
The link below shows typical configurations for up-flow non-backwashing carbon filter tanks. As you can see, the recommended media quantity for a 9" X 48" tank is 0.75 ft3. Without a backwashing valve, you are reliant on water flow to faucets and appliances to cause the media to backwash each time water is being used. Because that flow rate is not consistent, less media will be installed to reduce the potential for media loss into the home's plumbing system when the flow rate is greatest causing the media to be lifted high in the tank.
Although it may be configured for upflow, because your tank is stuffed full, the media cannot move, lift and reclassify so water flow will eventually follow the same path through the media (channel), causing the carbon along that path to become overused compared to the remaining media. This can quickly reduce the effectiveness of the carbon as a filter.
When periodically backwashing carbon media, the media will be redistributed (reclassified) which will eliminate any channels so the flow path will be refreshed. Carbon fines and other debris that enter within the supply water will be flushed out to drain during each backwash cycle as opposed to flowing into your home's plumbing as is now occuring.
Water softener resin is not a good filter media so some carbon fines will likely make their way through the resin to end up in your faucet aerators. Because a water softener utilizes a backwash cycle, I anticipate the majority of carbon fines that enter the resin will be back washed out to drain, but I suspect some may become caught in the softener's injector which may result in occasional brine draw issues.
You may want to consider installing cartridge sediment filters both before and after the carbon filter. The one after should prevent carbon particles from entering the softener, and the filter prior to prevent any sediment and debris from entering the carbon from the incoming supply.
https://view.publitas.com/impact-water-products/2018-catalog-final/page/32-33