Barnette's conjecture is an unsolved problem in graph theory. The problem states that every 3-regular (cubic), 3-connected, planar, bipartite (Barnette) graph is Hamiltonian. Partial results have been derived with restrictions on number of vertices, several properties of face-partitions and dual graphs of Barnette graphs while some studies focus just on structural characterizations of Barnette graphs. Noting that Spider web graphs are a subclass of Annular Decomposable Barnette (ADB graphs) graphs and are Hamiltonian, we study ADB graphs and their annular-connected subclass (ADB-AC graphs). We show that ADB-AC graphs can be generated from the smallest Barnette graph using recursive edge operations. We derive several conditions assuring the existence of Hamiltonian cycles in ADB-AC graphs without imposing restrictions on number of vertices, face size or any other constraints on the face partitions. We show that there can be two types of annuli in ADB-AC graphs, ring annuli and block annuli. Our main result is, ADB-AC graphs having non singular sequences of ring annuli are Hamiltonian.