Correct. The ONLY wallets that I know of that use a "wallet.dat" are Bitcoin Core and the many full node implementations that were created by forking the Bitcoin Core source code.
So far, you've received a lot of mediocre (or horrible) advice from sig ad users that care more about increasing their post count than actually helping.
What you need to do in order to back up your wallet will depend on the wallet that you are using. Each wallet has it's own backup requirements, and it is your responsibility to understand how to properly use whatever software you've chosen to use.
You've mentioned Mycelium. In that case, you only need to back up the seed words once. All the addresses and private keys that Mycelium generates are all generated from that set of seed words. You will be able to recover the entire wallet at any time you need to by using the seed words. You can store as many copies of those seed words as you like, but it is your responsibility to make sure of two things...
1. You will always be able to find a readable copy of the seed words when you need to.
2. Nobody else will have access to your seed words.
If you fail at responsibility #1, then you may find yourself in a situation where your bitcoins have not been taken, but you can no longer access them. They will be permanently lost to the entire bitcoin system.
If you fail at responsibility #2, then you may find yourself in a situation where someone else spends your bitcoins without your permission.
Note that, if you install a new Mycelium wallet and don't build it from a set of seed words, then it will generate a new set of seed words specifically for that wallet. In that case, you would have separate wallets each using Mycelium and each with their own set of seed words to keep track of.
Creating a backup of your wallet before trading in your phone takes care of responsibility #1. However, it does NOT take care of responsibility #2. There is a possibility that whomever receives your phone when you trade it in might try to extract your wallet from the phone without your knowledge or permission. There are a few possible ways to manage this risk.
1. Prior to trading in, you could create a temporary wallet on a different device (some other phone, or computer), and create a backup of it. Then create a transaction to send ALL of your bitcoins from your phone to this temporary wallet. Now when you trade in your phone. if someone succeeds in extracting your wallet from your phone, it won't matter since there aren't any bitcoins associated with that wallet any longer. Then after you set up new wallet on your new phone AND create a backup of that new wallet, you could send your bitcoins from your temporary wallet to this new phone wallet.
2. You could learn how to securely wipe all data from the phone before trading it in (the process may vary from phone to phone, so make sure you understand the proper process for your specific phone). This method is fast and easy. It also avoids the transaction fees of sending from your old wallet to a temp wallet, and then the transaction fees of sending from the temp wallet to a new wallet (assuming you set up your new wallet on the new phone by recovering from your seed words). However, there is a risk that the wipe won't be as secure as you think it is, or that you don't understand the proper process to implementing such a wipe.