Hi Mike. Yes, this is the correct cartridge assembly/motor for your 59896777 pump.
No, there is no replacement motor for the UP10-15B5 sku 59896253 so the pump would have to be replaced. This is an OEM pump so please consult the manufacturer of the unit this pump was supplied with for the correct replacement model.
Based on the model number provided it seems that sku 506173 is the correct replacement cartridge kit, not this one.
Normally if a pump is run dry, it is most likely no longer usable and should be replaced. It would be hard to determine if the pump is still operational.
Normally if a pump is run dry, it is most likely no longer usable and should be replaced. It would be hard to determine if the pump is still operational.
It is to hard to say for sure but there is a chance that by running it dry the pump is ruined. Try to get the air out of the pump and make sure the shaft is turning freely and if that doesn't fix the issue then the whole pump will probably need to be replaced.