Based on your descriptions, it is entirely acceptable to use the nonlinear state equations for simulation purposes. If your manually-defined Jacobian representation yields inaccurate responses, it is possible that the linearization components have been calculated incorrectly.
%% nonlinear state equations
function dx = ode(t, x)
dx(1) = x(2);
dx(2) = - x(2) - sin(x(1));
dx = [dx(1);
dx(2)];
end
%% simulation
[t, x] = ode45(@ode, [0 10], [1 0]);
plot(t, x), grid on
