Hi and welcome to the forum
They are probably stressed from being moved. Give them a week or two to recover and see how they are then.
Bigger water changes are better for fish. I like to do a 75% water change and gravel clean each week to reduce nutrients and disease organisms. Just make sure any new water is free of chlorine/ chloramine before it's added to the tank.
If you do a 25% water change each week you leave behind 75% of the bad stuff in the water.
If you do a 50% water change each week you leave behind 50% of the bad stuff in the water.
If you do a 75% water change each week you leave behind 25% of the bad stuff in the water.
You don't need a back up filter unless you live in a remote place and it takes a week or two for things to get to you. If the filter packs up (stops working) reduce feeding and do big daily water changes to keep ammonia levels down.
Besides what Aiken mentioned about releasing fish into the wild. It is usually illegal because of the risk of them introducing diseases into the wild or becoming an invasive species, and goldfish are an invasive species in some countries.
Your fish are both white (light coloured) and one appears to have long fins, which will reduce its movement speed. The light colouration will make them prime targets for predators because they will stand out really well against the dark substrate found in most rivers and lakes. Your fish would be eaten shortly after being released because of their colour and long fins.
If they are seen by a government worker in a creek or lake, they will be caught and euthanised for being an invasive species.
If you release your fish into the wild, you will most likely be sentencing them to an early grave. Just let them settle into their new home and enjoy having your fish back.