genealogical dictionary of new england, James Savage
https://archive.org/details/genealogicaldic04savarich/page/242/mode/2upsee attached image. very early Swinertons
Might have Battles also, didn't look.
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https://www.familysearch.org/en/tree/person/details/KZQS-C9G-----
PLYMOUTH CHURCH RECORDS VOLUME II
at
https://www.colonialsociety.org/node/302maybe him
Augst 11, 1751
Timothy the Son of James Swinnerton & wife
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There should be a lot of records on this family, at least there usually are for the early families in Salem, Plymouth, etc. There are a lot of people descended from them. The information might not all be on ancestry or familysearch, but, somewhere out there, there probably is someone's personal website with what you are looking for.
Plus a lot of records might be online elsewhere, like the Plymouth Church records above.
There is a series of books titled Vital Records of [name the city], Massachusetts. Try to find them.
edited to add, here are Salem and Plymouth:
Vital Records of Salem to the end of the year 1849
https://archive.org/details/vitalrecordsofsa03esse/page/n5/mode/2upfor Plymouth:
https://archive.org/details/vitalrecordsofpl00vana--------
For $$, if you want, try americanancestors.org. Check their Great Migration series also.