In the equation you provided, x(t) = Aexp(-at) cos(wt), the amplitude of the oscillatory component is indeed Aexp(-at), where A is the initial amplitude of the oscillation.
The term Aexp(-at) represents the decay of the amplitude over time due to the exponential term exp(-at). As time progresses, the exponential term decreases, causing the amplitude of the oscillation to decrease exponentially.
However, when we refer to the amplitude of x(t), we typically consider the maximum value of the oscillatory component, which is A. The cosine function, cos(wt), oscillates between -1 and +1, but the overall magnitude of the oscillation is determined by the amplitude A.
To clarify, Aexp(-at) represents the time-dependent decay of the amplitude, while A represents the maximum magnitude of the oscillation at any given time.