The Shapiro-Stiglitz model. Why under full employment do unemployed workers find jobs instantly?
I think that the confusion is caused by the different meaning of "full employment" in both contexts.

In the Shapiro-Stiglitz model, full employment in the sense that $\bar{L}\rightarrow LN$ implies that the job finding rate $a\rightarrow \infty$, which means no frictional unemployment.

In the other definition, full employment is merely defined by any remaining unemployment being only frictional (or structural/voluntary, not important here). Situations of no full employment could exist, for example, due to minimum wages, according to this definition. This view considers the unemployment arising from workers being laid off and waiting some time to find a new job (frictional unemployment) as a technological necessity that cannot be avoided so as to further increase efficiency.

Last but not least, please note that your quote from Romer concerns the labor supply curve only, while the employment rate will be determined in equilibrium.