Combination electric heat/a.c. unit works fine in "on" position. Fan switch works fine. Won't work in auto. White-Rodgers mercury style thermostat. Do I just need to replace thermostat?

When you say "auto", are you referring to the fan selector switch, or the mode selector switch? Do you have a voltmeter? If so, check for 24v from "G" to "C" at indoor unit with system set to heat/cool, fan set to auto. If other parts of system run but fan does not AND you do not measure 24v G-C, the fan circuit is not being powered. Check again with fan switch set to "on".

On the base of the thermostat, there are 2 choices for the fan: Auto and ON. I have the Fan set to Auto, and the system has 3 choices: Cool, Off, and Heat. I have that set to Heat.
When I take the cover off the thermostat, and have the Fan set to On, I can hear the furnace switch on and off as the mercury bulb bounces back and forth. But if I max out the dial to 80, and leave the fan set to Auto, nothing happens. Bad thermostat? I hope so.

Probably, but even if it isn't you should get rid of that mercury in favor of an electronic/digital type. Spend the few extra $$ for a good brand (Honeywell), avoid the cheapies (Hunter, Maple Chase). If you need help with the installation/wiring, you'll find it here as well. Good luck, we're all counting on you... :D