I have heard it suggested that convergent validity may still be adequate with AVE 0.6 (see p. 9 of Fornell and Larcker, 1981). You might consider testing AVE for each factor versus shared variance to explore discriminant validity, proposed by Fornell and Larcker (1981) and elaborated on by Farrell (2010).
You may also want to consider that typical cutoffs for fit indices differ for ULS vs. ML estimation (Xia & Lang, 2019).
I hope this helps!
Farrell (2010)
Article Insufficient Discriminant Validity: A Comment on Bove, Perva...