What's wrong with my gas furnace?

It is most likely the fan/limit switch as several other have suggested. From your description I don’t believe you have a high efficiency furnace requiring a separate draft inducer fan. I also doubt that a system that age has any electronic control boards. Inspect the settings on your fan/limit switch and make adjustments, but be prepared to replace it, they tend to get weak with age. It’s a fairly simple procedure but [disclaimer] should only be performed by a qualified service technician [/disclaimer]. Don’t poke around in it indiscriminately, it has 120 V power fed to it. Unplug the furnace before you adjust it, just to be safe.

One other thing, the heat anticipator in your thermostat (as noted previously) fools your thermostat into thinking it’s a little warmer than it really is. It does this because your thermostat is heavily influenced by the temperature of the wall it’s mounted on. If it waited until the wall was at the setpoint, the air in the room would be uncomfortably warm. A common setting for heat anticipators for gas furnaces is around 0.6. Smaller numbers will result in longer heating cycles, larger numbers in shorter cycles. Once set, it never needs to be adjusted. The anticipator has no direct connection with the fan. It only controls the gas valve (via your thermostat temperature). Your fan will will be controlled by the fan/limit switch, which is controlled by the temperature of the heat exchanger.