The enumeration of lozenge tilings of hexagons with holes has been studied intensively in recent years. Researchers tried to find shapes and positions of holes in hexagonal regions so that the number of lozenge tilings of the resulting regions is given by a simple product formula. In the present work, we consider new regions that are hybrids of regions studied by the first author (hexagons with intrusions) and Ciucu (F-cored hexagons). Then, we show that the tiling generating functions of these new regions under a certain weight are given by simple product formulas. To give a proof, we present shuffling theorems for lozenge tilings of hexagons with intrusions, which give simple relations between the tiling generating functions of two related hexagonal regions with intrusions.