The burning question - what make and model of Hayward sand filter (and pump) do you have? What leads you to decide that the filter and pump need to be replaced?
The pool turning green is purely a chemical problem. Taking an inordinately long time to clear the water AFTER doing a full SLAM might indicate filter issues. Properly sized sand filters can do the job well, hence wanting to know what you have now.
Going bigger/different with the pump and filter won't reduce or prevent the pool from going green. That only happens when you allow the amount of Cl needed for your CYA level to drop too low. Lets get into that part first - it may be you can save all the money, and put it toward something that may permanently solve the problem (like a SWCG).
I unknowingly did get a pool with my house that had a very undersized sand filter, and fought the algae battle for literally decades. But I also didn't know about the relationship between Cl and CYA. So the pool would go green, I would "shock" it, and then it was a issue to get the water back to clear as the filter was not up to the job. Many purchases of floc's and clarifiers. Then the green reappeared a week later, and the cycle started again.
Go to Settings over on the side here, and update your signature to list: pool gallons, type of construction (vinyl, plaster, etc), specific brand and models of all you equipment (pump, filter, heater and any other), and the model of the Taylor test kit you have. That way we don't have to keep asking.
List out the latest test result: FC, CC, pH, TA, CH, CYA, CH in a reply here.
Then we can have a better discussion about what to do...
