Paying only the required fee is enough in most cases but since the fees are also fluctuating you can sometimes end up in a situation such as the one in the OP.
That's why I suggested making RBF-enabled transaction. In this way, you don't have to pay fee more than required and you will be able to increase the fee, if needed.
As far as I'm aware RBF only works if you are the sender, so you can't use it while withdrawing BTC from a website for example.
There is no option to set the fee rate on custodial services. In custodial services, you usually pay a fixed fee and you aren't the one who set the transaction fee.