How to smoothly adjust sine amplitude
There's no problem letting the coefficients be a slowly varying function of time,

$$S(t)=a(t)\sin(t)+b(t),$$

for example a linear ramp 
$$a(t)=(a_1-a_0)\frac{t-t_0}{t_1-t_0}+a_0$$ for $t\in[t_0,t_1]$ and similar for $b(t)$.