Summary: Given a \(C^{*}\)-dynamical system \((A,G,\alpha)\), we say that \(A\) is a weakly proper \(X\rtimes G\)-algebra if there exists a proper \(G\)-space \(X\) together with a nondegenerate \(G\)-equivariant *-homomorphism \(\phi:C_0(X)\to\mathcal{M}(A)\). Weakly proper \(G\)-algebras form a large subclass of the class of proper \(G\)-algebras in the sense of Rieffel. In this paper, we show that weakly proper \(X\rtimes G\)-algebras allow the construction of full fixed-point algebras \(A^{G,\alpha}_u\) corresponding to the full crossed product \(A\rtimes_{\alpha}G\), thus solving, in this setting, a problem stated by M. A. Rieffel in his 1988 original article on proper actions [Mappings of operator algebras, Proc. Jap.-US Jt. Semin., Philadelphia/PA (USA) 1988, Prog. Math. 84, 141–182 (1991; Zbl 0749.22003)]. As an application, we obtain a general Landstad duality result for arbitrary coactions together with a new and functorial construction of maximalizations of coactions. The same methods also allow the construction of exotic generalized fixed-point algebras associated with crossed-product norms lying between the reduced and universal ones. Using these, we give complete answers to some questions on duality theory for exotic crossed products recently raised by Kaliszewski, Landstad, and Quigg.