Hot water coming out of cold
 

I have a Price Pfister shower faucet, it is a single lever turn handle
type. When I turn it to the cold water setting I get hot water for the
first few minutes. It then goes cold but it is very hard to regulate
warm water, I either get hot or cold. I was assuming it is a valve
within the cartridge that separates the hot and cold. I am also
getting hot water coming out of the sink tap for about 30 seconds now
as well. Will replacing the cartridge in the shower faucet solve both
problems or is something else going on with the plumbing?