New results are reported on the simulation of the rise of a deforming gas bubble surrounded by a liquid exhibiting a blend of shear-thinning and viscoelastic properties. A new hybrid model has been developed which combines the Carreau model for emulating the shear-thinning property of the liquid with the Chilcott-Rallison (FENE-CR) model for treating the viscoelastic property of the liquid. The new hybrid model allows one to independently prescribe the shear-thinning and viscoelastic properties (parameters) to be used in a simulation. Computations are implemented using the coupled level-set and volume-of-fluid (CLSVOF) method for tracking the deforming gas-liquid interface. The interfacial jump conditions are enforced with the sharp interface approach. In this study, four parameters, the Eotvos number, Carreau model parameter for slope of decreasing viscosity, Deborah number, and viscoelastic polymer dumbbell length parameter, are varied in order to gain understanding for gas bubble morphology in liquids exhibiting both shear-thinning and viscoelastic properties. It is found that the extent of bubble deformation and the liquid stress field are strongly dependent on these 4 parameters with minimal correlation between the shear-thinning and viscoelastic parameters. The ability to vary shear-thinning and viscoelastic parameters independently of one another in the new hybrid Carreau and FENE-CR model enables one to conveniently predict gas bubble behavior in complex liquid polymers by parameterizing the new model from basic measurements of the polymers.