The solution I was looking for ended up being more simple than I had thought. After trying a number of combinations to creating a control to override the time value, it turns out that that the Start Time Offset parameter in the Geometry Cache does exactly what I was looking for. 
At some point, I am going to try to figure out how to override that with a variable, that way I could range map a bunch of these things to one variable, but one thing at a time. Thanks again for your help!