If I remember correctly, Mplus gives you Chi-square-difference tests between models when running EFA. Those are very informative for choosing the best model.

If you are concerned about there being common variance in addition to specific variances, I would recommend looking into bifactor models (of which second-order models are a restrictive special case).

To Philipp K. Masur , Gregory Camilli , & Heiko Breitsohl these are all incredibly thoughtful and helpful responses to my question. The poor model fit of the 1-factor model seems to suggest other sources of common variance exist. I shall proceed with bi-factor EFA and then use mulidimensional IRT modeling (bi-factor IRT).

You might want to try confirmatory factor analysis with a single latent variable but with correlated measurement errors, and compare that to a 2-factor model (with or without correlated errors, while estimating the correlation between the two latent factors). See Bollen and Grandjean, American Sociological Review 1981, for an example of this approach.

Burke D. Grandjean , thank you for the insight, and quick question. If I were to run a unidimensional graded response model with correlated residuals, would that violate local dependence (an assumption of IRT)? Is there a way to address correlated residuals in IRT?

As far as I know, IRT usually employs pretty strong assumptions. I tend to use CFA instead, so I can pick and choose which assumptions to employ as constraints, and which to relax and estimate. So I'm inclined to think CFA with correlated measurement errors is an alternative to an IRT approach, but maybe there's a way to combine the two. Wish I could be more help on that question.

Thanks much, Burke D. Grandjean !

0.70 is acceptable