You do not need to sum them up. Form 1040 and 1040-NR are two separate forms. You just need to mail them together to the IRS. It will be just $X and $Y and IRS will send you the total refund amount.
If you decided to file as dual status, for year 2020, you will file a Form 1040-NR for the period when you are a nonresident, only report US-sourced income. For the period when you are a resident, you will file a Form 1040 reporting worldwide income. As TurboTax does not support nonresident tax forms, you are suggested to visit Sprintax for the Form 1040NR portion. You will then use TurboTax to complete the resident portion. You cannot e-file. Print out all the tax forms and paper file to the IRS.
For instructions how to file Dual status, see DualStatus under "What and Where To File for a Dual-Status Year" and 1040NR.
To read more about the pros and cons on dual status, click here: Dual status
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