When I test my circuit, I see much more drift at the output than I would expect based on the offset voltage drift spec. Can you help me understand where this additional drift might be coming from?

LT6370 is an excellent choice for this application.  Instrumentation amplifiers which use an external resistor to set their gain do so in conjunction with other resistors that are integrated inside the amplifier.  The temperature coefficient of these internal resistors is unlikely to match that of the external resistor used to set the gain.  The result is a gain drift, which you may be interpreting as an offset drift because you see the output moving with temperature.  The max gain drift spec of the amplifier accounts for the internal resistors only and assumes ideal external resistors.  In practice the TC of the external resistors is additive with the internal resistors and must be minimized through component selection.  The error budget in the LT6370 datasheet for this application tabulates this error contribution, along with many others to aid in optimizing you design.