I think you'll find that your heatsink will need to be replaced. The copper pipes actually contain a gas to help the heat travel along them; over time this gas can leak, rendering your heatsink useless. You can tell if this is the case by opening up your laptop and touching the copper pipe while your laptop is on - if it's cold to the touch, you'll need to replace it. Feeling cold air coming out the vent while the laptop is overheating would also indicate this.
I wouldn't bother trying to use copper shims. I spent at least $30 trying to avoid replacing the heatsink in my laptop, when I should have spent that on a new heatsink to begin with.
You can easily order a replacement heatsink from China using a website such as aliexpress.