The recent success in applying integrability-based methods to study examples of gauge/gravity dualities in highly (super)symmetric settings motivates the question of whether such methods can be carried over to more physical and less symmetric cases. In this thesis we consider two such examples of string sigma-models, which interpolate between integrable or solvable limits. First we consider classical string motion on curved p-brane backgrounds for which the sigma-model interpolates between the integrable flat space and AdS(k)xS(k) coset or WZW sigma-models. We find that while the equations for particle (i.e. geodesic) motion are integrable in these backgrounds, the equations for extended string motion are not. The second example we consider is string theory on AdS3xS3xT4 with mixed Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) 3-form fluxes, which interpolates between the integrable pure R-R and the pure NS-NS theory that can be solved using CFT methods. The dispersion relation and S-matrix for world-sheet excitations, which are the essential ingredients in solving for the string spectrum, are only partially fixed by integrability and symmetry arguments. By constructing the mixed flux generalisation of the dyonic giant magnon soliton, which we show can be interpreted as a bound-state of excitations, we determine the dispersion relation for massive excitations. We also construct the mixed flux generalisation of the folded string on AdS3xS1 and show that, at leading order in large angular momentum on AdS3, its energy is given by the pure R-R expression with the string tension rescaled by the R-R flux coefficient. Further, we derive the bound-state S-matrix and its 1-loop correction by considering the scattering of dyonic giant magnons and plane waves. From this we deduce the semiclassical and 1-loop dressing phases in the massive sector S-matrix, which we find to agree with recent proposals.