Accelerometer readings
The equation for the accelerometer's output voltage at time 't' is,

$$
y(t) = k\, a(t)
\;\;\;\;\;\;\;\;\;\;(1)
$$

where

y(t) := The accelerometer's output voltage at time 't'
k    := A conversion constant with units of V*s^2/m (see eq. 7 below)
a(t) := The acceleration at time 't' with SI units of m/s^2


You need velocity, not acceleration. We know that acceleration at time 't' is the first derivative of velocity at time 't' taken with respect to time 't':

$$
a(t) = \frac{dv(t)}{dt}
\;\;\;\;\;\;\;\;\;\;(2)
$$

Substitute equation (2) into equation (1) and using separation of variables solve for velocity at time 't', v(t):

$$
y(t) = k\, a(t)=k\,\frac{dv(t)}{dt}
\;\;\;\;\;\;\;\;\;\;(3)
\\[0.2in]
\rightarrow
dv(t)=\frac{1}{k}\,y(t)\,dt
\;\;\;\;\;\;\;\;\;\;(4)
\\[0.2in]
\rightarrow
\int_{v(t_0)}^{v(t)}dv(t)=\frac{1}{k}\int_{t_0}^{t}y(t)\,dt
\;\;\;\;\;\;\;\;\;\;(5)
\\[0.2in]
\rightarrow
v(t)=\frac{1}{k}\int_{t_0}^{t}y(t)\,dt+v(t_0)
\;\;\;\;\;\;\;\;\;\;(6)
$$

where the integral \$\int y(t)dt\$ has units of \$Volts \cdot seconds\$.

In other words, you'll need to integrate the accelerometer's output voltage \$y(t)\$ from time \$t_0\$ to \$t\$, divide that result by the conversion constant 'k', and then add in the accelerometer's initial velocity \$v(t_0)\$ at starting time \$t_0\$ to get the velocity at time 't', \$v(t)\$.

Solving for the value of the conversion constant 'k' is straightforward. For example, if the accelerometer's output voltage is \$1\,mV\$ for an applied acceleration of \$1\,g\$, then,

$$
k=\frac{1\,mV}{9.807\,m/s^2} \bigg\rvert \frac{1\,m}{100\,cm} \bigg\rvert\frac{2.54\,cm}{1\,inch}=\frac{2.59\,\mu V}{in/s^2}
\;\;\;\;\;\;\;\;\;\;(7)
$$


  I see a dampened sine wave on my scope as the output of the charge amplifier.


If the damped sinewave signal has exponential decay, then the formula for \$y(t)\$ in equation (6) is of the form shown in equation (8):

$$
y(t)=A\,e^{-\lambda\,t}\,sin(2\pi f t + \phi)
\;\;\;\;\;\;\;\;\;\;(8)
$$

where


&lambda; := Decay constant (1/s)
A := Sine wave's undamped amplitude
f := Sine wave's frequency (1/s)
&phi; := Sine wave's starting phase angle (radians)