Bridport, Dorset.has a number of Swaisie's in the parish register,
I see a John Swaisie son of Robert baptised there in 1621 but I looked at the image for the 1619 page and saw no John son of John The register is on ancestry, I suggest a careful look (many libraries have a subscription) might be useful
Notice the profile also suggests Sherborne. That's a bigger parish, no baptism of a John Swaisie or variant indexed but worth checking.
NB noticed that your date in the question is earlier than the profile linked to. The Bridport registers only seem to go back to 1600.