But I have to ask, why not just send the bitcoin to your new address? Is it a smallish amount, or are the fees looking too daunting?
Because you can't send bitcoins with a paper wallet. It's just a piece of paper.
If you have say a desktop wallet with some coins and you also have some coins on a paper wallet, can you sweep those coins on the paper wallet to the desktop wallet or will that replace your whole wallet and balance on the desktop wallet?
People that write "wallet address" are probably unaware that wallet can contain more than one address. There is no danger of losing bitcoins by importing or sweeping a private key.
Some wallets let you
import a private key. In that case, the wallet assumes control of the private key and its address.
Many wallets don't have that capability because they don't store private keys. These wallets may offer the ability to
sweep your private key, rather than
import it. The difference is that when a wallet sweeps a private key, it creates a transaction that sends the bitcoins from the swept address to itself and then forgets the swept private key.
Sweeping a private key is generally safer because when you take the paper wallet out of protective storage, you increase the risk of exposing the private key to someone. Once the bitcoins are swept from the paper wallet, the swept private key no longer has any bitcoins. The downside to sweeping a private key is that the bitcoins have to be sent and that will cost money and time.