The opinion of the House of Shammai, as quoted in the Mishnah (Gittin 90a), is:
Beit Shammai say: A man may not divorce his wife unless he finds out about her having engaged in a matter of forbidden sexual intercourse [devar erva], as it is stated: “Because he has found some unseemly matter [ervat davar] in her, and he writes her a scroll of severance” (Deuteronomy 24:1).
(edited from Sefaria translation)
If her husband did divorce her for less than adultery, the divorce is still valid, even though he wasn't allowed to have divorced her, as is clear from this discussion (on the same page):
Rav Pappa said to Rava: What if the husband found about her neither forbidden sexual intercourse nor any other matter, but divorced her anyway?
Rava said to him that the answer can be derived from what God reveals in the Torah with regard to a rapist: “He may not send her away all his days” (Deuteronomy 22:29), indicating that all his days he remains obligated to remarry her. Only there the husband is obligated to remarry her, as God reveals as much. But here, what he did, he did.
(Even though Rashi says the question is about the opinion of the House of Hillel, the answer is equally valid for the House of Shammai.)
So she wouldn't be committing adultery if she remarried after being divorced for something other than adultery. Even though a woman who committed adultery remains forbidden to the person with whom she committed adultery even after the divorce (Sotah 5:1), the divorce allows her to marry anyone else.