No, you do not need to amend.
The payments were an advance credit for 2020. 2019 information was used in sending out the payments since 2020 tax returns had not been filed at the point the stimulus payments were sent out. They are being reconciled when you do your 2020 return. In some cases, people may have been given a stimulus payment based on their 2019 dependents that did not qualify based on their 2020 dependents, and vice versa.
In your particular case, you received a stimulus for 1 dependent based on your 2019 return. Now that you are filing your 2020 return, you still only have 1 dependent and you received the stimulus already for 1 dependent. If you were claiming the other child still in 2020, you would have 2 dependents and would qualify for the rebate recovery credit.
As far as your fiance, from what you have said, he is claiming 1 dependent this year. In 2019, did he have any dependents on his return? How much stimulus money did he receive? As long as you are showing the child as a qualified dependent, he should get the stimulus for himself and 1 dependent, based on his 2020 return.
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