The water pressure in the shower only works when you use the hot, on cold it just trickles, plus we just put in a new cartridge.

Sounds like you have some debris from the pipe walls blocking the cold water inlet port of the cartridge. When you had the valve apart did you turn the water back on for a minute to flush the supplies?
Turn the water off to the valve and remove the cartridge. Checking clean the inlet ports and this time flush the supplies before you reassemble.
Let me know if that helps. Good luck, Tom

Is it a single handle faucet ?