This paper investigates a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity(⋆){ut=D1uxx−χ1(uk1vx)x+μ1u(λ1−u+a1v),vt=D2vxx+χ2(vk2ux)x+μ2v(λ2−v−a2u), under homogeneous boundary conditions of Neumann type for u and v, in an open bounded interval Ω⊂R, where Di,χi,λi,ai>0 and μi≥0 for i∈{1,2}. In Tao and Winkler (2021) [1]; (2022) [2], Tao and Winkler studied the system (⋆) with k1=k2=1 and obtained the global existence and asymptotic behavior. We study the system (⋆) with k1,k2≠1 and prove that:•If k1∈(1,4031) and k2∈[14,12−9k18−5k1), then the model (⋆) possesses global weak solutions for arbitrarily large positive initial data in H1(Ω).•If k1∈(1,1110) and k2∈[23,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1=μ2=0 stabilizes toward homogeneous equilibria (1|Ω|∫Ωu0,1|Ω|∫Ωv0).•If k1∈(1,98) and k2∈[12,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1,μ2>0 converges to constant stable steady state (λ1,0) provided that λ2≤a2λ1 and the tactic sensitivities χ1 and χ2 are suitably small.