The wallet is built on an HD (or hierarchical deterministic) framework, which has a different method for address generation and management. Each public address your wallet generates stems from your wallet's xPub (or Extended Public Key). Once your public address receives an incoming payment, a new address will automatically be generated and display when you click on receive.
If you use the same address each time you receive funds, it becomes easy for anyone to track your entire payment history. This method of address generation improves privacy by automatically presenting you with a new address when you’re expecting payment.
There are wallets that only switch 10 Bitcoin addresses at random (each one individually in its wallet), while other wallets generate a completely new Bitcoin address for each transaction. This is totally normal and it depends on the type and the settings of the Bitcoin wallet. These addresses are not going to become out of date, you can still receive Bitcoin for an earlier Bitcoin address, even though your wallet has since generated a new address.
It is definitely worth to use such a wallet that generates to each transaction a completely new Bitcoin address, therefore ensuring a higher level of private security. You do not need to deal with it separately, most of the wallets handle this automatically.
As many wallets generate a completely new address after each transaction, it is highly important and recommended to double check the address you have given to us, because cryptocurrency transactions are basically irreversible, therefore coins sent to badly-entered addresses cannot be repayed!