How would I measure the Carbon Dioxide content in Coca Cola over a period of time?
Measure the change in mass over time of the remaining liquid. Though some water will also evaporate, you can control for that by keeping the humidity near 100%.

If you have to be precise, collect the outgassed $\ce{CO2}$ in a liquid-nitrogen cold trap. Check the mass of the condensate, which should equal that lost from the soda. Check purity, to be really finicky.