Hello Sanjana,
Hope you are doin well !
To answer your query, YES you can give CAT 2022, and get admission in an IIM (depending on your percentile) as well as XAT 2023.
As, you will be a graduate in August 2023, you can give CAT, and XAT and start your B-school from the same year of graduation.
Hope my answer was helpful. Please let me know in case of any other queries.
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