If my thermostat's been set to 64 or 65 all day and I bump it up to, say,
68, I find maybe 20 minutes later that the house temp (measured at the
thermostat) is still where it was before raising the setting. Sometimes a
bit lower, because I also find that the furnace is blowing cold air. But
before I raise the setting, the temp shown on the thermostat matches the
lower temp it was set at before. So, I know that while I was away, the
system was maintaining the lower temp correctly.

I've only noticed this for the past couple of days, and I've been too busy
when arriving home to check & see if the furnace blew ANY warm air in the
first few minutes after raising the temp. When it's functioning correctly,
there's warm air within 30 seconds of the furnace turning on. With the
partial information I've provided, does this sound more like a thermostat
issue or a problem with the furnace?

The thermostat is an Aprilaire 8363 (digital, controls furnace and AC).
Installed by reputable contractor. The gas furnace is a 12 year old Goodman
installed by a contractor hired by previous homeowner. It's been serviced by
my choice of contractor for the past 8 years.