Deacon John Shaw, referred to in Vol 3, page 23 of the History of Weymouth appears to refer to a different person than John Shaw who married Alice Phillips. The Deacon John Shaw is shown in Volume 4, page 613, and appears to be a sibling of Elizabeth. Deacon John Shaw is not shown with a daughter named Elizabeth.
Torrey, in New England Marriages prior to 1700, page 2258 says Edward Bates married: "Edward (1655-) & Elizabeth [?EDSELL]; by 1679/[80?]; Weymouth {Weymouth 3:23; Bates Bulletin 1913:19, 4 ser. 154; Bates (n.d.) 20; Sv. 1:138}"
Torrey, in the same reference, page 26166 has this to say about Joseph Poole: "Joseph (-1706) & 1/wf Elizabeth [SHAW] (1656-), m/2 Peter HOLBROOK, m/3 Robert WEARE; by 1674; Weymouth {Weymouth 4:475; Poole 9; Gen. Adv. 3:128, 4:64; Bassett-Preston 215, 254; Reg. 59:328; Backus Anc. 155, 156}". The timing of these second and third marriages for Elizabeth seem suspect though.
No records of either marriages have been found in Massachusetts Vital Records to 1850, Weymouth volume, although both marriages appear (from Torrey) to have occurred there. Broadening the search to all of Massachusetts turns up no records for either marriage in that database.
Since Elizabeth couldn't have been married and having children with two different men at the same time, and since her father, John Shaw's will provides evidence of her marrying Poole, it seems that she should be disconnected from Edward Bates as a spouse, and create a new Elizabeth Unknown as Edward Bates' wife, to which the children would be connected. A Disputed Origins section could be created containing the speculation about Edward being her husband.
In online genealogy forums, this question has come up, and the postings that I have read said there is no primary source data to show Elizabeth (daughter of John and Alice Shaw) married Edward Bates. It seems that the researcher in History of Weymouth errored with respect to Edward Bates and Elizabeth, and a lot of other researchers relied on this erroneous information, perpetuating the myth.